The Assured Value Machine

The Assured Value Machine (AVM) is a pricing function that maps token supply to token price and acts as the counterparty to every trade. The design specifies no order book, no liquidity pool, and no external market maker. A buy pays capital into a reserve and mints supply. A sell burns supply and draws capital out of that same reserve. Because buyers create the reserve, no outside party has to pledge liquidity, and no outside party can withdraw it and strand the market.

Definition

The curve y(x) has three segments.

        | f       0  <= x <= x1    (floor)
y(x) =  | R(x)    x1 <  x <  x2    (ramp)
        | M(x)    x2 <= x          (main)

Floor. A constant at height f. This is the guaranteed minimum redemption price for every token in circulation. The floor can only increase.

Ramp. A strictly increasing function on the open interval (x1, x2). The ramp is the transition zone between the floor guarantee and the open market, and it is steeper than the main schedule.

Main. A strictly increasing function on [x2, inf). This is the long-run pricing schedule, where a market spends most of its life.

The curve is continuous at both boundaries, so R(x1) = f and R(x2) = M(x2). Neither boundary is a free parameter. x1 is wherever the ramp meets the floor and x2 is wherever the ramp meets the main; both move as a consequence of changes to the floor or to either schedule. At launch the two boundaries coincide, the ramp has zero width, and the entire curve is the main schedule anchored at the initial floor price. The ramp comes into existence only when a floor raise separates the boundaries.

The ramp constraint

Every conforming implementation must satisfy one structural property:

R(x) < M(x)   for all x in (x1, x2)

Both segments increase, and they meet at x2, but the ramp approaches that meeting point from below. Extrapolated backward through the transition zone, the main schedule sits higher than the ramp everywhere. The gap between the two is an area deficit, and that deficit is the budget that funds a floor raise. Remove the constraint and there is nothing to redistribute, so an area-preserving raise becomes impossible. A single uniform schedule cannot support one. Two schedules, a steep ramp and a gentle main, are the minimum structure the mechanism requires.

The linear implementation

The current implementation instantiates both R and M as straight lines, with the ramp slope fixed steeper than the main slope at market creation. Both slopes hold for the life of the market; the floor and the two schedule positions carry all the state that changes.

At launch the two boundaries coincide, which reduces the curve to a single ray rising from the initial floor price. A ramp segment appears only once a floor raise separates them.

Linearity is what keeps the machine cheap to run. Every integral is a trapezoid, so fills, contraction updates and floor-raise solutions are all closed-form arithmetic on a handful of values.

Reserve and solvency

The reserve is the capital backing all tokens in circulation. It equals the area under the price curve from zero to current supply.

reserve(x) = integral from 0 to x of y(s) ds

Solvency follows from that definition rather than from monitoring. Buys deposit exactly the area they traverse; sells withdraw exactly the area they retrace. The equality between reserve and area is preserved by construction at every step, which makes it an identity rather than a condition. There is no collateral ratio to watch, no oracle to poll, and no liquidation engine held in reserve against a shortfall. A state in which the reserve cannot redeem every token at the price the curve assigns is not reachable through issuance and redemption.

Buying and selling

A buy moves supply from x to a higher x'. The buyer pays the integral of the curve across the new region, and that capital enters the reserve. The fill price is that integral divided by x' - x, which for a linear segment is the midpoint of the starting and ending price. A sell moves supply down and pays out the integral across the region it retraces.

Neither operation reshapes the curve while supply stays at or above x2. Trades move along the schedule; they do not move the schedule.

Contraction

A sell that pushes supply below x2 forces the curve to adapt. Two cases exist.

Selling into the ramp, with the new supply at or above x1, leaves the ramp untouched and shifts the main schedule down until it meets the ramp at the new supply. The boundary x2 retreats to that point and the ramp narrows.

Selling into the floor, with the new supply below x1, collapses the ramp to zero width. Both schedules re-anchor at the new supply at height f, so x1 and x2 land on the same point. If supply returns all the way to zero, the curve resumes its launch shape, though at whatever floor prior raises have established.

Contraction is permanent with respect to buying. Later buys walk up the shifted curve and do not restore the old main schedule. Only a floor raise advances x2 again. The floor itself never moves during contraction, in either case.

Guaranteed and discovery regions

The area under the curve decomposes into two parts, and the split is the clearest way to read what the machine is doing.

The guaranteed region is the rectangle from zero to supply at height f, with area f * x. Every token redeems for at least f no matter what other participants do. This portion of the reserve is unconditional.

The discovery region is everything above f, between x1 and current supply. It is the integral of y(s) - f across the ramp and main segments. All price sensitivity lives here. Demand creates this area and sells destroy it. The market price of a token is f plus the height of the curve above the floor at current supply, so in the floor region the premium is zero.

Raising the floor converts discovery-region area into guaranteed-region area. Total reserve stays the same. The fraction of it that is unconditional goes up.

Sensitivity by segment

The slope of the active segment is the market’s price sensitivity to supply changes.

Segment	Slope	Sensitivity
Floor	0	None. Sells against the floor have no price impact, so the floor behaves as an infinitely deep bid.
Ramp	m1	High. Price moves quickly through the transition zone.
Main	m2	Moderate. The regime the design targets for long-run stability.

For a buy of dx inside a single linear segment of slope m, price moves by exactly m * dx and the fill is the midpoint between start and end price. A buy that crosses a boundary sums the contributions of each segment it touches.

Cost of price movement

On one linear segment starting at price f with slope m, the capital needed to double the price is:

K = 3 * f^2 / (2 * m)

and after a total capital inflow of C the price multiple is:

multiple = sqrt(1 + 3*C/K)

Price therefore moves with the square root of capital inflow, and capital scales with the square of the multiple. This is a geometric consequence of the linear segment and holds at any magnitude of f. It also explains the two-slope design: the ramp’s steeper slope gives it a smaller K than the main schedule, so the same inflow moves price further there.

Example

Take a market whose floor is 0.10 and whose spot price is 0.19, with supply sitting on the main schedule some distance past the ramp-main boundary. That last condition is what an area-preserving raise requires.

Of the 0.19 a token is worth, the first 0.10 is guaranteed area: it is redeemable at the floor regardless of what anyone else does. The remaining 0.09 is discovery area, which exists only while demand holds it up. A floor raise moves value from the second figure into the first without changing their sum.

Related

• Floor raising rearranges area to lift f without touching the reserve or the spot price.
• Cash advance draws liquidity against the guaranteed region.
• Call options mint at the floor and translate both schedules rightward.