Key Formulas Cheat Sheet

Keep this page handy when you need the CLMSR math without re-reading the entire specification. Each block lists the formula, explains when it applies, and points to the contracts that enforce it.

Potential, prices, and loss bound

• Potential: $C(q) = \alpha \ln \left( \sum_b e^{q_b / \alpha} \right)$ governs the entire surface. The share inventory $q_b$ and liquidity parameter $\alpha$ live in WAD precision.
• Bin price: $p_b = e^{q_b / \alpha} / \sum_j e^{q_j / \alpha}$. Because every bin shares the same potential, the prices always sum to 1.
• Maker loss bound: $\text{Loss}_{\max} = \alpha \ln(n)$ where $n$ is the number of bins. Adjusting the tick spacing or liquidity parameter alters the bound predictably.

Trading a range

• Weight update: buying $\delta$ shares multiplies the weights inside each bin of the range by $\varphi = e^{\delta / \alpha}; selling flips the sign of$\delta$.
• Cost: $\Delta C = \alpha \ln(\Sigma_{\text{after}} / \Sigma_{\text{before}})$, where the sigma terms are the summed weights before and after applying $\varphi$.
• Chunking rule: exponents are evaluated in chunks no larger than MAX_EXP_INPUT_WAD = 1e18, keeping (chunk/alpha) <= 1.

Outcome and tick mapping

• Tick index: $b = \mathrm{clamp}(\lfloor (\text{OutcomeRaw} - L)/s \rfloor, 0, n-1)$.
• Bin interval: [L + b s, L + (b+1) s) with the upper edge open to prevent overlap.

Precision and rounding

• Currency to WAD: multiply SUSD amounts by 10^12; divide by the same factor to convert back.
• Rounding policy:
  • Buys and increases: round up (fromWadRoundUp).
  • Sells, decreases, closes, and payouts: round down (fromWadFloor).
  • Recommended minimum trade size: $\delta_{\min} = 0.01$ SUSD (UI enforced until the Solidity guard ships).

Guard rails

• MAX_EXP_INPUT_WAD = 1.0e18
• MIN_FACTOR = 0.01e18, MAX_FACTOR = 100e18
• MAX_TICK_COUNT = 1_000_000
• MAX_CHUNKS_PER_TX = 1_000

Need the longer story? Step back to the Mechanism Overview or jump into the detailed discussions in Cost Function & Rounding and Safety Bounds & Parameters.