???? Mathematical Formulas
1. Marginal Minting Price (from Supply)
P_mint($L) = P(0) × ((β + $L) / β)^α
Calculates the marginal price to mint the next token based on the current token supply $L.
This represents the instantaneous price at a given supply level.
2. Marginal Minting Price (from Deposit)
P_mint(Q) = P(0) × [(α+1) × Q / (P(0) × β) + 1]^(α/(α+1))
Calculates the marginal price after depositing Q amount of external currency.
This shows what the price will be after the deposit is processed.
3. Tokens Minted from Deposit
Δ$L = β × [(α+1) × Q / (P(0) × β) + 1]^(1/(α+1)) - β
Calculates how many tokens are minted when depositing Q currency, starting from zero supply.
This is the inverse of the cost function.
4. Cost to Mint Tokens
Q = P(0) × β / (α+1) × [((β + Δ$L) / β)^(α+1) - 1]
Calculates the total cost to mint Δ$L tokens starting from zero supply.
This represents the area under the bonding curve.
5. Incremental Cost (with existing supply)
Cost(Δ$L, $L₀) = P(0) × β / (α+1) × [((β + $L₀ + Δ$L) / β)^(α+1) - ((β + $L₀) / β)^(α+1)]
Calculates the cost to mint Δ$L additional tokens when $L₀ tokens already exist.
Used for incremental minting scenarios.
6. Average Minting Price
P_average($L) = P(0) × β / ((α+1) × $L) × [((β + $L) / β)^(α+1) - 1]
Calculates the average price paid per token when minting $L tokens from zero.
Always less than the marginal price due to the convex curve shape.
Parameters:
P(0) = 1 USDC (Initial price, fixed)
α (alpha) = Curve steepness coefficient (0 < α < 1)
β (beta) = Virtual supply coefficient (0 < β < 1, but typically in thousands)
$L = Native token symbol
Q = External currency (USDC) deposit amount
α (alpha) = Curve steepness coefficient (0 < α < 1)
β (beta) = Virtual supply coefficient (0 < β < 1, but typically in thousands)
$L = Native token symbol
Q = External currency (USDC) deposit amount
⚙️ Simulation Parameters
Actual β = displayed value × 1000