> ## Documentation Index
> Fetch the complete documentation index at: https://docs.hoodstar.fun/llms.txt
> Use this file to discover all available pages before exploring further.

# The Bonding Curve

> The exact math behind hoodstar.fun pricing — constant-product curve, virtual reserves, and graduation numbers.

Every hoodstar.fun token trades on the same **constant-product bonding curve** until it graduates. The curve is an automated market maker with no order book: price is purely a function of reserve state, so it's fully deterministic and quotable in advance.

## The formula

The curve maintains two *virtual reserves* and keeps their product constant:

```text theme={null}
virtualEth × virtualToken = k
```

* Buys add ETH to `virtualEth` and remove tokens from `virtualToken` → price rises.
* Sells do the opposite → price falls.
* The spot price at any moment is `virtualEth / virtualToken`.

"Virtual" means the reserves start pre-seeded so the token has a sensible non-zero starting price with no initial liquidity deposit. The virtual ETH offset also guarantees the curve always holds enough real ETH to buy back every circulating token.

## Curve parameters

| Parameter                             | Value                                    |
| ------------------------------------- | ---------------------------------------- |
| Total token supply                    | 1,000,000,000 (fixed)                    |
| Initial virtual token reserve         | 1,073,000,000                            |
| Supply sold on the curve              | 793,100,000 (79.31%)                     |
| Supply reserved for Uniswap liquidity | 206,900,000 (20.69%)                     |
| Initial virtual ETH reserve           | 1.6 ETH (launch parameter)               |
| Trade fee                             | 1% in ETH (0.7% protocol / 0.3% creator) |

With the current 1.6 ETH virtual reserve, that works out to:

| Milestone                           | Approximate value                            |
| ----------------------------------- | -------------------------------------------- |
| Starting price                      | \~0.0000000015 ETH per token (FDV ≈ 1.5 ETH) |
| ETH raised at graduation            | \~4.53 ETH net (≈ 2.83× the virtual reserve) |
| Price appreciation across the curve | \~14.7×                                      |
| FDV at graduation                   | \~21.9 ETH                                   |

<Info>
  The virtual ETH reserve is the protocol's single economic knob and is fixed per deployment — it scales the ETH cost of the curve without changing its shape. All other constants are hard-coded in the contract.
</Info>

## Buys, sells, and rounding

For a buy of `ethIn` (after the 1% fee is skimmed):

```text theme={null}
tokensOut = virtualToken − k / (virtualEth + ethInAfterFee)
```

Sells are the mirror image. All rounding is done **in the pool's favor** (`k` can never decrease), which protects the curve's solvency — this invariant is fuzz-tested in the contract suite.

Quotes from the contract's `quoteBuy` / `quoteSell` functions are exact. The [SDK](/developers/sdk) also ships a bit-exact client-side mirror of this math so UIs and bots can quote without an RPC round-trip.

## Graduation trigger

The curve tracks how much of the 793.1M sale allocation remains. The buy that empties it:

1. Fills exactly to the remaining supply (excess ETH is refunded in the same transaction),
2. Marks the curve **graduated** — permanently frozen, and
3. Makes the token eligible for [migration to Uniswap v3](/protocol/graduation).

The graduation threshold is **supply-based, not market-cap-based** — there's no oracle and nothing to manipulate: sell out the curve and it graduates, every time.

## Why sells can never break the curve

The invariant `realEth = virtualEth − virtualEth₀` holds at all times: the real ETH held by the contract is exactly what buyers put in (net of fees), which is exactly what the curve would owe if every holder sold back down to the starting point. The curve is always solvent by construction.