> For the complete documentation index, see [llms.txt](https://docs.fira.money/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.fira.money/developers/protocol-contracts/core/market-math-core.md).

# MarketMathCore

MarketMathCore encodes the time-decaying pricing curve and trade execution math used by FiraMarket. OracleLib provides the TWAP ring buffer.

{% hint style="info" %}
**Source:** `src/fira_bonding/core/Market/MarketMathCore.sol`, `src/fira_bonding/core/Market/OracleLib.sol`
{% endhint %}

## MarketMathCore

This library operates entirely on in-memory `MarketState` structs for gas efficiency. It never reads from or writes to storage directly.

### Pricing model

The AMM prices BT relative to FW using a logit curve:

```
exchangeRate = ln(proportion / (1 - proportion)) / rateScalar + rateAnchor
```

Where `proportion = totalBt / (totalBt + totalAsset)` and `totalAsset` is FW reserves converted to underlying asset units via the BC index.

* **`rateScalar`** scales inversely with time-to-expiry (`scalarRoot × 365 days / timeToExpiry`), making the curve steeper near maturity.
* **`rateAnchor`** is recalculated after each trade so that the current pool proportion yields the last recorded implied rate.

### Liquidity math

* **First deposit:** Mints `sqrt(fw × bt) - MINIMUM_LIQUIDITY` LP tokens, locking 1000 wei permanently.
* **Subsequent deposits:** Mints proportional to the limiting token.
* **Removal:** Returns proportional shares of both reserves.

### Trade execution

`executeTradeCore` loads pre-computed values, calculates the FW amount from the exchange rate, applies fees, and updates state. Fees are split between the swapper (as worse pricing) and the reserve (as a percentage of the fee). The post-trade implied rate is recalculated and stored.

### Key invariants

* Exchange rate is always >= 1e18 (reverts otherwise)
* BT proportion cannot exceed `MAX_MARKET_PROPORTION` (96%)
* Implied rate is always > 0 after a trade
* `MINIMUM_LIQUIDITY` prevents the first-depositor attack

## OracleLib

Adapted from Uniswap V3's oracle. Stores `Observation` structs (timestamp + cumulative `ln(impliedRate)` + initialized flag) in a fixed-size array of 65,535 slots.

* **`write()`** — Appends a new observation if the block timestamp differs from the last
* **`grow()`** — Pre-initializes storage slots to reduce gas during swaps
* **`observe()`** — Returns cumulative values at requested time offsets using binary search
* **TWAP** is computed by the caller as `(cumulative[now] - cumulative[ago]) / duration`


---

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