Bonding Curve - [~^.

Overview

402.cat uses a constant product bonding curve (similar to Uniswap and Pump.fun) to provide instant liquidity for newly created tokens. Tokens trade on the curve until graduation, when they automatically move to Uniswap V4.

The Formula

The bonding curve follows the constant product formula:

k = virtual_usdc × virtual_tokens

Where k is a constant that never changes (the invariant).

Buy Formula

When you buy tokens, you add USDC to the curve and remove tokens:

tokens_out = virtual_tokens - (k / (virtual_usdc + usdc_in))

Sell Formula

When you sell tokens, you add tokens to the curve and remove USDC:

usdc_out = virtual_usdc - (k / (virtual_tokens + tokens_in))

Virtual vs Real Reserves

402.cat implements a dual reserve system (Pump.fun model):

Virtual Reserves (For Pricing)

Used to calculate prices via the constant product formula:

virtual_usdc: Virtual USDC reserve
- Initial: 7,403,872 ($7.40)
- Increases on buys, decreases on sells
virtual_tokens: Virtual token reserve
- Initial: 1,073,025 tokens (1,073,025,000,000,000,000,000,000 with 18 decimals)
- Decreases on buys, increases on sells

Real Reserves (Actual Holdings)

Tracks actual tradeable assets:

real_usdc: Actual USDC deposited
- Initial: 0
- Accumulates from net buys (after fees)
- Used for LP creation at graduation
real_tokens: Tokens available for purchase
- Initial: 793,100 tokens (79.31% of total supply)
- Decreases on buys, increases on sells
- Graduation triggers when this reaches 0

Key Insight: Virtual reserves determine price via the formula, while real reserves track what’s actually available for trading.

Initial Conditions

For a 1,000,000 token supply on testnet:

Parameter	Value	Notes
Total Supply	1,000,000 tokens	Fixed at creation
Real Tokens	793,100 tokens	79.31% available for trading
LP Reserve	206,900 tokens	20.69% reserved for liquidity
Virtual USDC	7,403,872 ($7.40)	Initial virtual reserve
Virtual Tokens	1,073,025 tokens	Initial virtual reserve
Real USDC	$0	Accumulates from trades
Initial Market Cap	$6.90	Starting valuation
Initial Price	~$0.0000069	Starting token price

The constant k is calculated as:

k = virtual_usdc × virtual_tokens
k = 7,403,872 × 1,073,025,000,000,000,000,000,000
k ≈ 7.94 × 10^30

Virtual Token Derivation

Virtual tokens are derived mathematically to ensure the graduation price matches the LP price:

T_v = T_r² / (2×T_r - total_supply)

Where:

T_v = virtual_tokens
T_r = real_tokens (793,100)
total_supply = 1,000,000

This ensures smooth price continuity between the bonding curve and the Uniswap V4 pool after graduation.

Price Calculation

Current Price (Spot Price)

The current price per token is:

price = (virtual_usdc / 10^6) / (virtual_tokens / 10^18)

Result is in USDC per token.

Market Cap

Market cap is the current price multiplied by total supply:

market_cap = price × (total_supply / 10^18)

Graduation Progress

Track how close a token is to graduation:

progress = (initial_real_tokens - current_real_tokens) / initial_real_tokens

Returns 0.0 to 1.0 (0% to 100%).

Buy and Sell Mechanics

Buy Operation

Calculate fee: fee = usdc_in × 1% = usdc_in × 100 / 10000
Net USDC: net_usdc = usdc_in - fee

Calculate tokens using virtual reserves:

tokens_out = virtual_tokens - (k / (virtual_usdc + net_usdc))

Check availability: Ensure tokens_out ≤ real_tokens
Update reserves:
- virtual_usdc += net_usdc
- virtual_tokens -= tokens_out
- real_usdc += net_usdc
- real_tokens -= tokens_out

Sell Operation

Calculate gross USDC using virtual reserves:

gross_usdc = virtual_usdc - (k / (virtual_tokens + tokens_in))

Check availability: Ensure gross_usdc ≤ real_usdc
Calculate fee: fee = gross_usdc × 1% = gross_usdc × 100 / 10000
Net USDC to user: net_usdc = gross_usdc - fee
Update reserves:
- virtual_usdc -= gross_usdc
- virtual_tokens += tokens_in
- real_usdc -= gross_usdc
- real_tokens += tokens_in

Atomic Execution

All buy/sell operations use PostgreSQL row-level locking (FOR UPDATE) to prevent race conditions during concurrent trades. This ensures the bonding curve state remains consistent even under high-frequency trading.

Fee Structure

Bonding Curve Phase

Buy Fee: 1% (100 basis points)
- Added to the payment amount
- Example: Buy $0.10 worth = pay$ 0.101 total
Sell Fee: 1% (100 basis points)
- Deducted from proceeds
- Example: Sell for $0.10 = receive$ 0.099

Fees go to the platform and are used for:

Platform operations
LP token distribution at graduation
Referral rewards (10% of fees)

Post-Graduation (Uniswap V4)

After graduation, trades happen on Uniswap V4 with a custom hook:

Hook Fee: ~0.3% (30 basis points)
Fee Distribution:
- 33% to token creator
- 33% to platform
- 33% to LP holders

Graduation Mechanism

Trigger

Graduation occurs when all real tokens are sold:

real_tokens == 0

At this point:

All 793,100 tradeable tokens have been purchased
Market cap has reached the graduation threshold
The bonding curve is “complete”

Testnet vs Mainnet

Network	Initial Mcap	Graduation Threshold	Total Supply
Base Sepolia	$6.90	~$69	1,000,000
Base	$6,900	~$69,000	1,000,000

What Happens at Graduation

Airdrop: Remaining LP reserve tokens (206,900) are airdropped to existing holders proportionally
🔥 CAT Buy Pressure: ALL accumulated real_usdc is used to buy the platform token (CAT) from the market
- This creates automatic buy pressure on CAT with every graduation!
- Example: $69,000 graduation =$ 69,000 CAT market buy 🚀
- More successful tokens = more CAT demand
LP Creation: Uniswap V4 Token/CAT pool is created with:
- Purchased CAT tokens (from step 2)
- LP reserve tokens (206,900)
LP Locking: 80% of LP tokens are locked forever
Creator Reward: 20% of LP tokens go to the token creator
Hook Deployment: Cat402Hook is attached for fee distribution

Platform Token Economics: Every successful token that graduates creates buy pressure on CAT! The first graduated token BECOMES the CAT platform token. All subsequent graduations must buy CAT to create their liquidity pools. This makes CAT increasingly valuable as the platform grows. 💰

See Graduation for detailed CAT mechanics and platform token economics.

Price Impact

The constant product formula creates automatic price slippage - larger trades have higher price impact:

Example: Buy Impact

Buying $1 USDC worth when virtual_usdc = 7,403,872:

Small buy ($0.10): ~0.00135% price impact
Medium buy ($1.00): ~0.0135% price impact
Large buy ($10.00): ~0.135% price impact

Price impact increases non-linearly with trade size.

No Additional Slippage Parameters

Unlike DEXes, the bonding curve has no slippage tolerance settings. The constant product formula automatically determines the execution price based on the trade size and current reserves.

Comparison to Pump.fun

402.cat’s bonding curve is modeled after Pump.fun with key similarities:

Feature	402.cat	Pump.fun
Formula	Constant product (x×y=k)	Constant product (x×y=k)
Reserves	Virtual + Real separation	Virtual + Real separation
Graduation	To Uniswap V4	To Raydium
LP Distribution	80% locked, 20% creator	Similar distribution
Fee Model	1% buy/sell	1% buy/sell

Key Differences:

Network: Base (Ethereum L2) vs Solana
Graduation DEX: Uniswap V4 vs Raydium
Hook System: Custom V4 hooks for fees vs Raydium standard
Payment Protocol: x402 micropayments vs SOL gas fees

Advanced Topics

Invariant Preservation

The constant k must remain unchanged throughout trading:

k_before = virtual_usdc_before × virtual_tokens_before
k_after = virtual_usdc_after × virtual_tokens_after

// This must be true:
k_before == k_after

Any deviation indicates a bug in the implementation.

Price Continuity

The virtual reserve design ensures price continuity at graduation:

bonding_curve_price = virtual_usdc / virtual_tokens
graduation_lp_price = real_usdc / lp_tokens

// These should be approximately equal
bonding_curve_price ≈ graduation_lp_price

This prevents arbitrage opportunities during the graduation transition.

Edge Cases

Last Tokens Available:

If a buy would exceed real_tokens, the buy is capped at available tokens
User receives whatever is left at the calculated price
Transaction succeeds, token graduates immediately

Insufficient Liquidity on Sells:

If gross_usdc > real_usdc, the sell fails
This can happen if many people sell after a large buy
Rare in practice due to balanced in/out flows

Monitoring the Curve

Key Metrics to Watch

Graduation Progress: How close to selling out

progress = 1 - (real_tokens / initial_real_tokens)

Price Trend: Is the price increasing or decreasing

price_change = (current_price - previous_price) / previous_price

Volume: Total USDC traded
```
volume = real_usdc + total_fees
```
Holder Distribution: Who owns the tokens
- Check via positions endpoint
- High concentration = risky
- Distributed = healthier

API Endpoints

Check bonding curve state via:

# Get full bonding curve state
GET /tokens/:id

# Returns:
{
  "bondingCurve": {
    "virtualUsdc": "7500000",
    "virtualTokens": "1070000...",
    "realUsdc": "150000",
    "realTokens": "920000...",
    "k": "8025000..."
  }
}

Next Steps

Graduation

Learn about the graduation process

Fee Structure

Understand all fees in the system

Token Lifecycle

Complete journey from creation to graduation

API Reference

Build applications on the bonding curve

The bonding curve is deterministic - given the same reserves, the price is always the same. This makes it predictable and fair for all participants.

Getting Started

httpcat-cli

MCP Server

Core Concepts

​Overview

​The Formula

​Buy Formula

​Sell Formula

​Virtual vs Real Reserves

​Virtual Reserves (For Pricing)

​Real Reserves (Actual Holdings)

​Initial Conditions

​Virtual Token Derivation

​Price Calculation

​Current Price (Spot Price)

​Market Cap

​Graduation Progress

​Buy and Sell Mechanics

​Buy Operation

​Sell Operation

​Atomic Execution

​Fee Structure

​Bonding Curve Phase

​Post-Graduation (Uniswap V4)

​Graduation Mechanism

​Trigger

​Testnet vs Mainnet

​What Happens at Graduation

​Price Impact

​Example: Buy Impact

​No Additional Slippage Parameters

​Comparison to Pump.fun

​Advanced Topics

​Invariant Preservation

​Price Continuity

​Edge Cases

​Monitoring the Curve

​Key Metrics to Watch

​API Endpoints

​Next Steps

Graduation

Fee Structure

Token Lifecycle

API Reference

Overview

The Formula

Buy Formula

Sell Formula

Virtual vs Real Reserves

Virtual Reserves (For Pricing)

Real Reserves (Actual Holdings)

Initial Conditions

Virtual Token Derivation

Price Calculation

Current Price (Spot Price)

Market Cap

Graduation Progress

Buy and Sell Mechanics

Buy Operation

Sell Operation

Atomic Execution

Fee Structure

Bonding Curve Phase

Post-Graduation (Uniswap V4)

Graduation Mechanism

Trigger

Testnet vs Mainnet

What Happens at Graduation

Price Impact

Example: Buy Impact

No Additional Slippage Parameters

Comparison to Pump.fun

Advanced Topics

Invariant Preservation

Price Continuity

Edge Cases

Monitoring the Curve

Key Metrics to Watch

API Endpoints

Next Steps