This page contains the mathematical foundations and mechanics that underpin the system.
Most users do not need to read this section — it is intended for advanced users, integrators, and researchers.

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Position Value Tracking

A leveraged position grows faster than its collateral. The relationship is: δV∗V∗=L⋅δVcVc\frac{\delta V_*}{V_*} = L \cdot \frac{\delta V_c}{V_c} Where:
  • V∗V_* = value of the leveraged position
  • VcV_c = collateral value
  • LL = compounding leverage factor
Integrating: V∗∝(Vc)LV_* \propto (V_c)^L Example: If L=2L = 2 and the liquidity pool price is pLP=pp_{LP} = \sqrt{p}, then: V∗∝(p)2=pV_* \propto \left(\sqrt{p}\right)^2 = p This means the position tracks the underlying asset’s price exactly — while still earning fees.

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APR Approximation

The expected return of the system can be approximated as: APR≈2rpool−(rborrow+rloss)APR \approx 2r_{pool} - (r_{borrow} + r_{loss}) Where:
  • rpoolr_{pool} = yield from the base pool
  • rborrowr_{borrow} = borrowing cost
  • rlossr_{loss} = losses from rebalancing
APR is positive as long as the amplified pool yield exceeds costs.

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Releverage Algorithm

To maintain constant leverage LL, the debt must stay aligned with collateral and oracle price pop_o: d~=L−1L po y~\tilde{d} = \frac{L-1}{L} \, p_o \, \tilde{y} where y~\tilde{y} is the collateral amount at oracle parity. The system uses a modified AMM invariant: (x0(po)−d) y=I(po)(x_0(p_o) - d)\,y = I(p_o) Solving for x0(po)x_0(p_o) leads to a quadratic expression: x0(po)=poy+po2y2−4 poyd(L2L−1)22(L2L−1)2x_0(p_o) = \frac{p_o y + \sqrt{p_o^2 y^2 - 4\,p_o y d \left(\tfrac{L}{2L-1}\right)^2}}{2 \left(\tfrac{L}{2L-1}\right)^2} At parity (p=pop = p_o), the value of the pool is: V=x02L−1V = \frac{x_0}{2L - 1} This formula is also used when calculating shares for deposits and withdrawals.

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Deposits

When a user deposits:
  1. They provide cryptocurrency amount cinc_{in} and specify debt increment Δd\Delta d.
  2. The system borrows Δd\Delta d stablecoins and adds (Δd,cin)(\Delta d, c_{in}) to the pool, receiving LP tokens ll.
  3. New state variables (d+Δd,y+l)(d + \Delta d, y + l) are used to compute updated x0x_0 and pool value VV.
  4. Synthetic tokens are minted to the user proportional to the increase in VV.

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Withdrawals

For a withdrawal of tt LP tokens (from total supply ss):
  • Collateral and debt are reduced by t/st/s.
  • Withdrawn LP tokens are redeemed such that stablecoins exactly match the reduced debt δd\delta d.
  • Any remaining crypto is returned to the user.
This ensures fairness: no one can “cherry-pick” debt/collateral ratios at exit.

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Arbitrage & Virtual Pool

Arbitrageurs interact via a virtual pool that combines:
  • The releverage AMM,
  • The underlying liquidity pool,
  • Flash loans for stablecoins.
For stable → crypto swaps, the required flash amount ϕ\phi satisfies: ϕ2+ϕ(x0−d+x−ry)−yxr=0\phi^2 + \phi \big(x_0 - d + x - ry\big) - yx r = 0 with: r=xcSc(1−f),D=b2+4yxr,b=x0−d+x−ryr = \frac{x_c}{S_c}(1 - f), \quad D = b^2 + 4yxr, \quad b = x_0 - d + x - ry and solution: ϕ=12(D−b)\phi = \tfrac{1}{2} \big(\sqrt{D} - b\big) Where:
  • ff = fee of the releverage AMM
  • dd = debt
  • yy = collateral amount
  • x0x_0 = oracle-derived state variable
  • xc,yc,Scx_c, y_c, S_c = reserves and supply in the base pool

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Summary

  • Value growth follows V∗∝(Vc)LV_* \propto (V_c)^L
  • APR depends on pool yield vs. borrow + rebalance costs
  • Constant leverage is maintained by a dedicated releverage AMM
  • Deposits and withdrawals adjust both collateral and debt proportionally
  • Arbitrageurs ensure smooth operation via the virtual pool