How it works

This page follows the Orbital paper (Paradigm, 2025) and shows where each idea lives in the program.

One point on a sphere

A pool holds n coins (2 to 16, MIN_ASSETS and MAX_ASSETS). Its reserves are a vector x = (x_1, ..., x_n). The pool keeps x on a sphere of radius r centred at (r, r, ..., r):

sum_i (r - x_i)^2 = r^2

No reserve can go below zero: the farthest a point on the sphere can be from the centre in any one direction is r.

Prices

The price of coin i in units of coin j is the slope of the sphere at the current point:

price(i in j) = (r - x_i) / (r - x_j)

A coin the pool holds a lot of is cheap; a coin it holds little of is expensive. When every reserve is equal, every price is 1. That is the equal price point, where every stablecoin sits when all trade at $1.

Bands are caps

A band is the set of points on the sphere within a fixed distance of the equal price point. Its edge is where the sphere meets a plane at right angles to the line from the centre through the equal price point. Seen on the sphere, it is a cap.

The width of a band follows from the price at which a single falling coin reaches its edge. For three coins drawn in 3D, those widths are:

A band never needs to hold more of a coin than its edge allows, so a liquidity provider deposits only what the band can actually reach. The paper calls the rest virtual reserves. For five coins, the paper gives about 15x capital efficiency for a $0.90 band and about 150x for $0.99.

Interior and pinned bands

At any moment each band is either:

• interior: the pool's point is inside its cap, and it trades every coin, or
• pinned (Band.boundary == 1): the point has reached its edge. The band stops taking the falling coin and keeps trading the others along its rim.

The program stores every band's geometry in Band.beta, gamma, mu, sigma (Q64.64 fixed point) and its total radius in Band.radius. Interior bands combine into one sphere; pinned bands combine into a smaller sphere orthogonal to the equal price direction. The paper calls the combined shape a torus. Pool.tau stores where the pool sits between band edges.

A swap

swap(coin_in, coin_out, amount_in, min_out):

1. Takes the fee: fee = ceil(amount_in * 200 / 1,000,000). 25% of it goes to Pool.protocol_fees, the rest is credited to bands by fee weight.
2. Solves the trade on the combined shape for the net input (math::trade).
3. If the trade reaches a band edge, it splits there, moves that band between interior and pinned, and continues. The number of crossings is reported in the Swapped event.
4. Checks out >= min_out, moves the tokens, and asserts each vault still holds what the pool owes.

Rounding

Every rounding step favours the pool: deposits round up what you pay, withdrawals round down what you get, fees round up.