Bonding curve

A bonding curve is a market maker that holds a launch's supply from the first second and always quotes a price. Nobody sets that price. It follows from how much of the supply has been bought.

#How the price works

Eden uses a constant-product curve with a virtual (phantom) reserve of the pair asset. The phantom reserve sets the opening price, so the first buyer does not get the supply for nothing. It is never withdrawable and never paid out.

Spot price at any moment is $p=(\Phi +Q)/Y$. It opens at $p_0=\Phi /S$.

Buying with $\Delta Q$ of the pair asset: the trading fee $\tau$ and any snipe tax are taken from the input first, and the rest is priced on the curve:

Selling $\Delta Y$ tokens: the curve prices the trade, then takes the fee from what you receive:

Rounding is always in the curve's favour, so the invariant $(\Phi +Q)\cdot Y\ge \Phi \cdot S$ holds after every trade.

#Every launch on the same terms graduates the same way

The phantom reserve is always one quarter of the graduation threshold ($\Phi =G/4$). That fixes three things for every launch, whatever its pair asset:

For stock pairs the threshold is set from the stock's price when the pair was approved. It is re-priced only for new launches, and only if the stock has moved more than 5 %. A launch's terms never change after it is created.

#Worked example (ETH pair)

At the opening, $\Phi +Q=1.25$ ETH and $Y=10^9$. A buyer spends 0.5 ETH after the snipe window:

1. The fee is taken from the 0.5 ETH, leaving $\Delta Q^{\prime }=0.485$ ETH.
2. Tokens out $=0.485\times 10^9/(1.25+0.485)$ ≈ 279.5 M, about 28 % of supply.
3. The spot price rises from $1.25\times 10^{-9}$ to about $2.4\times 10^{-9}$ ETH.

A second buyer spending the same 0.5 ETH gets fewer tokens, because the price has already moved. That is the curve doing its job: early risk is rewarded, and nobody can get a large position at the opening price.

#Price impact and slippage

Large buys move the price more than small ones, so the average price you pay on a large order is worse than the spot price you were quoted. Every trade carries a minimum-output bound you sign. If the price moves past it before your trade lands, the trade reverts and you keep your funds.

#When you cannot sell to the curve

Selling is always open, with one exception: once the curve has sold out, it holds exactly the reserves the pool is about to be built from. Letting anyone sell into them would change the pool every launch is meant to arrive at. Selling reopens in the pool after graduation, which normally happens in the same transaction that sold the curve out. If it did not, anyone can complete graduation themselves and trade the pool.

#Fees

The trading fee is charged in the pair asset, never in the launch token. You pay it on what you spend when you buy, and on what you receive when you sell. Creators can never add their own tax on top.

The full derivation, including liquidity growth after graduation, is in Scoring and rewards math.