The Sigma Invariant

The Sigma Invariant is the core mechanism powering Sigma Money, adapted from the original f(x) Protocol design. It enables on-chain leverage while preserving capital efficiency for decentralized stablecoins.

The Sigma Invariant ensures that the total value of all bnbUSD, combined with the total value of all xPOSITIONs and sPOSITIONs, always equals the total value of the protocol’s collateral reserves within the trading functions. By dynamically adjusting the leverage ratios of xPOSITION and sPOSITION alongside the peg ratio of bnbUSD, the system achieves seamless coordination between low-volatility stablecoins and high-leverage trading instruments.

The Sigma Invariant is:

$$(\ n\ -\ n^-\ )\ \times\ p\ =\ (\ b\ -\ b^-\ )\ +\ X\ +\ S$$

Where:

• n - the number of TOKEN collateral

• n^- - the number of TOKEN collateral borrowed to open short positions

• p - the TOKEN price in USD

• b - the number of bnbUSD minted from opening long positions

• b^- - the number of bnbUSD locked from opening short positions

• X - the NAV of all xPOSITIONs in USD

• S - the NAV of all sPOSITIONs in USD

This mechanism keeps the protocol balanced, solvent, and stable across market conditions, ensuring that every asset — whether stable or leveraged — remains fully backed by collateral on-chain.