Curve math

The forever curve is a constant-product book with a virtual ETH reserve.

Let

• P = phantom quote (default 1 ETH)
• R = real ETH reserve (starts at 0)
• T = token reserve (starts at total supply)
• V = P + R = virtual quote

Invariant: V * T is conserved across trades.

Buy

Spend q ETH:

tokensOut = T * q / (V + q)
R' = R + q
T' = T - tokensOut

Sell

Sell t tokens:

quoteOut = V * t / (T + t)

quoteOut cannot exceed R. Selling the exact inventory bought in a single prior buy returns that ETH (minus rounding toward zero).

Spot

price = V / T

Reported on-chain as spotPrice = V * 1e18 / T.

There is no protocol fee term in either formula.