The Bonding Curve

How to make it go brrr?

Bonding curves establish a clear relationship between token supply and price. As more tokens are bought or sold in the bonding curve, the price per token adjusts according to a predefined mathematical function. This provides clarity and predictability for the user.

Printr’s chain-agnostic bonding curve establishes a clear relationship between supply and price, independent of the specific blockchain.

Printr Bonding Curve Formula

The formula for the curve is:

$$Constant Product: k = (v + r) × t$$

where:

• v = virtual reserve = (T₀ × P₀) / PRECISION where P₀ = initial price

• r = actual reserve (ETH/WETH raised)

• t = token reserve (unsold tokens)

• T₀ = initial token reserve for curve

• PRECISION = 10^18

Price: P = (PRECISION × v × T₀) / t² (quadratic increase)

Buy Cost: cost = k/(t - Δt) - v - r Key: Initial price sets virtual reserve size, which anchors the entire curve. Higher P₀ → higher v → higher starting prices throughout.

Properties of the Curve

• Supply-Driven Price Increases: Prices rise as users mint more tokens, following the supply curve.

• Liquidity Smoothing: The virtual reserve a reduces sudden price jumps when users add new tokens.

• Supply Constraints: The function only works within valid supply limits, maintaining ecosystem stability.

The expression provides us a reliable pricing model for printed tokens while providing stability through the token's lifecycle.

Implications

The implications of using this bonding curve expression are:

• Once the specified token supply is sold, the contract graduates the token and moves it to the DEX by creating a liquidity pool.

• Each chain operates independently, meaning tokens may graduate at different times.

• If a token is deployed across multiple chains, its supply is split evenly.

• The price volatility is higher based on demand due to the reduced per-chain supply.

---

Common Questions

Since the bonding is chain-agnostic, what happens when a token graduates on a chain faster than others?

• Nothing exceptional really happens. The token on that chain becomes tradable on DEXs but is still tradable through our contract as it routes trades to DEXs under the hood.

• Graduation on one chain might trigger graduation on the other as a chain reaction. If a token is graduated, it means its price is higher. So it's profitable to buy it where it hasn't graduated yet, bridge, and sell where it's graduated already.

• This model's effect is cross-chain liquidity flow for the token. Price consistency is an important aspect of liquidity unification, and our chain-agnostic bonding curve model would help achieve that.