The bonding curve

A brand-new token has no market. The classic answers - the creator seeds a pool, or a market maker is hired - both put someone's capital in first and both invite rugs. A bonding curve removes the someone: the contract quotes a deterministic price at every moment from day zero, liquidity is never anyone's to withdraw, and the price path depends only on what people actually buy. The trade-off is that the curve phase is a one-venue market with a hard cap - which is what graduation resolves.

The curve is a constant product - the same x · y = k that powers Uniswap v2 - run on virtual reserves: bookkeeping numbers initialised at launch rather than deposited capital.

The spot price at any moment is y / x. Because y₁ = 3·y₀ is fixed by the contract, every curve has the same shape regardless of the supply or opening market cap a creator chose. Three consequences follow directly:

• The raise is twice the opening reserve. Net buying of y₁ − y₀ = 2·y₀ completes the curve (4 ETH at the default y₀ = 2 ETH).
• The curve sells ~73.3% of supply. Tokens sold at completion are x₀ − x₁ = ⅔ · x₀ = 73.33% of supply; the remaining ~26.7% is reserved for the graduation pool.
• Graduation price is 9× the opening price. y₁/x₁ = 3y₀ / (x₀/3) = 9 · y₀/x₀ - so a token that opens at a $2.5K market cap graduates around $22.5K, before whatever the open market does next.

Buy. The swap fee comes off your ETH first; the remainder moves the curve. Tokens out for a net input Δy:

tokensOut = x − k / (y + Δy)

Sell. The mirror image - tokens go in, ETH comes out along the same curve, and the fee comes off the ETH you receive:

ethOut = ( y − k / (x + tokensIn) ) · (1 − fee)

Buying pushes x down and y up, so the price rises; selling does the opposite. The round trip is symmetric apart from fees: nobody, including the platform, can extract value from the curve itself - a property the contract's test suite fuzzes rather than assumes. You can never sell more tokens than the curve has actually sold, so the curve always holds the ETH it owes.

The buy that would push y past y₁ is charged only what the curve still needs (plus fee on that amount); the excess is refunded in the same transaction. The token's state flips to Completed, trading on the curve stops, and the token waits - briefly - for the graduation keeper. No value is conjured at the boundary: the last buyer pays exactly the curve price for exactly the remaining tokens.

Supply

1,000,000,000

x₀

1,100,000,000

y₀

2 ETH

y₁

6 ETH

x₁

366,666,667

Raise

4 ETH

• Opening price y₀/x₀ ≈ 0.0000000018 ETH - an opening market cap of supply · y₀/x₀ = y₀/1.1 ≈ 1.82 ETH.
• After 1 ETH of net buys: y = 3, x = k/3 ≈ 733M - about 367M tokens sold and the price up 2.25× from open.
• At completion: y = 6, x ≈ 367M - 733M tokens (73.3% of supply) sold for a net 4 ETH, price 9× open.
• Into the pool at graduation: the 4 ETH raised plus the ~267M reserved tokens.

• Minimum buy: 0.0001 ETH - keeps dust from spamming the event stream.
• Max wallet: 2% of supply per wallet, when the launch kept anti-sniper on. Enforced on the buyer's post-buy balance, only during bonding.
• Slippage protection: every buy and sell carries a minimum-out bound derived from a fresh quote at submit time; if the price moved past your tolerance the transaction reverts instead of filling badly.
• No expiry. A curve that never fills simply keeps trading - there is no deadline after which funds are stranded or swept.

Curve-phase price action is real price action, so it deserves real charts. The token page charts every trade from the first one. Additionally, each launch deploys a small read-only companion contract that mirrors the curve's reserves in the exact shape DEX screeners already understand (a Uniswap-v2-style pair), so third-party chart sites can index the bonding phase too - most launchpads' tokens only appear on screeners after graduation.