Arbitrage and liquidity: the study
What the two loops earn and what a liquidity provider on Uniswap earns — with the arithmetic, on the white paper’s defaults and Arbitrum’s costs.
Notation and costs
Let B be backing per token, p_m the mint price, P the market price, f_r = 0.9 % the redemption fee, m = 1/(1 − drawdown) its multiplier, f_t the transfer fee (default zero), φ the pool fee tier, g the gas of one transaction, and s the slippage of a trade of the size in question. The corridor is a ≤ P ≤ b with a = B·(1 − f_r·m) and b = p_m.
Costs on Arbitrum: g is of the order of cents; φ is 0.05, 0.3 or 1 % on Uniswap v3, and on v4 whatever the pool’s creator set, fixed or dynamic; s depends on the pool’s depth. Uniswap v4 has been live on Arbitrum since January 2025, and since July 2026 a protocol fee is taken from a share of LP fees on v4 pools — it does not change the trader’s cost, only the provider’s take.
The upper loop: mint, close, sell
The loop is profitable when the market pays more than a fresh token costs to make: P·(1 − φ) − s − g > p_m. Minting creates a position at p_m; closing it is free and raises backing, because p_m > B by construction; selling on the pool pays φ and slippage. Profit per token: P·(1 − φ) − p_m − s − g/t, where t is the size.
Every round of the loop adds t tokens at p_m to the pool’s reserve — it is the saver’s upward channel, run by an arbitrageur. The loop stops when P falls to p_m/(1 − φ) + s; that is the effective ceiling, a little above the mint price. Since p_m rises without limit, the ceiling is a moving line, not a wall.
This is also the entry that reopens after saturation. When the market has cleared and P sits at the floor while p_m has kept rising, the loop is not profitable — the gap is wide the wrong way. It becomes profitable the moment buyers return and lift P above p_m/(1 − φ); the first minter after a drought closes above a backing that has not fallen, and the staircase resumes.
Figure 1. Two loops at two edges The upper loop mints, closes and sells when the price exceeds the effective ceiling; the lower buys and redeems when it falls below the effective floor. Both return the price into the band.
The lower loop: buy, redeem
The loop is profitable when the reserve pays more than the market asks: B·(1 − f_r·m) − P·(1 + φ) − s − g/t > 0. Buying on the pool pays φ and slippage; redeeming pays the fee f_r times the multiplier m; the payout is the reserve’s share.
The effective floor is a/(1 + φ) − s, a little below a. In a drawdown m > 1 and the floor moves down: at 38.2 % drawdown m = 1.618 and the fee is 1.46 %; at 61.8 %, 2.36 %. This is deliberate — a band that widens smoothly prevents a moment before which it pays to hurry out — and it means the lower loop is least active exactly when backing is furthest from its record.
Redemption is neutral, so the loop leaves B unchanged and the fee behind. It removes tokens from circulation; the provider’s pool gets thinner in ASTRX and richer in the stablecoin, which is what a range position does near its lower edge anyway.
The realised band
Between the loops the price is free, but not free of traders. A trader who buys at P_low and sells at P_high pays κ = 2φ·P + f_t·P + 2s + 2g/t per token. Profit is (P_high − P_low) − κ; traders act while it is positive and stop when it is not; so the realised band settles at w* ≈ κ. With g in cents and t in thousands of tokens, gas vanishes; with a deep pool, slippage is small; the band is set by the fee tier and the transfer fee.
At φ = 0.3 % and f_t = 0: w* ≈ 0.6 % plus slippage — under one per cent. At φ = 1 %: about 2 %. At φ = 0.05 %: a tenth of a per cent, but a tenth-of-a-per-cent pool attracts no liquidity for a new token. The tier is a choice between a tight band with thin liquidity and a wide band with a paid provider; the study’s recommendation is 0.3 % for launch and a v4 dynamic tier later, if a hook is written for it.
Figure 2. The realised band against the fee tier w* ≈ 2φ + f_t + slippage; gas is negligible on Arbitrum. The three v3 tiers and a dynamic v4 tier as a range.
The provider’s position in a bounded range
A concentrated-liquidity position with liquidity L on [a, b] holds, at price P inside the range, x = L·(1/√P − 1/√b) tokens and y = L·(√P − √a) stablecoin. At P = a it is all tokens; at P = b, all stablecoin. Its value is y + P·x. The capital needed to open it at P₀ is that value at P₀.
Impermanent loss is the difference between the position’s value and the value of the same capital held unchanged. For a range as wide as the corridor — b/a of order 1.5 to 2 — the loss at either edge is a few per cent of the capital. It is bounded because P cannot leave [a, b]: arbitrage at both edges returns it. The provider’s worst case is to be all-ASTRX at the floor — an asset with a defended floor — or all-stablecoin at the ceiling, having sold every token at the highest price the corridor allows.
Fee income is φ times the volume that passes through the range while the price is inside it, times the provider’s share of the range’s liquidity. Volume comes from the two loops and from traders; both loops are active when the corridor is wide and the market moves, which is the inflow phase.
Figure 3. A range position inside the corridor Value of a concentrated position against the price inside [a, b], compared with holding the same capital. The loss is bounded because the price is.
Three phases, with numbers
Assumptions for the table: a pool of 200 thousand dollars of two-sided liquidity in a 0.3 % tier, the provider holding one tenth of it; a corridor with b/a = 1.6; a transfer fee of zero. Volumes are illustrative: what changes between phases is where volume comes from, and the table shows the mechanism, not a forecast.
Inflow: daily volume of the order of the pool — traders, the upper loop, closings being sold. Fee income to the whole range: about 0.3 % of that a day, roughly 100 % a year on the pool’s capital; the provider’s tenth accordingly. Impermanent loss small: the price sits mid-to-high in the range and the range is moved up with the corridor every few weeks.
Stagnation: volume a tenth of the pool a day. Fee income about 10 % a year. The price drifts towards the floor; the position turns into ASTRX; the provider re-ranges downward-heavy or waits — the floor is defended, so waiting is not ruin.
Outflow: volume from the lower loop and from holders leaving — a fifth of the pool a day for a while, then less. Fee income between the two cases while it lasts. The position ends all-ASTRX at the floor, which is exactly the asset the next inflow phase will buy first.
The re-ranging cost is gas — cents — plus the realised difference between the old range’s holdings and the new range’s; on a corridor that rises by rule, re-ranging upward means selling some ASTRX into the new, higher range, which is the same act as taking profit.
Figure 4. Fee income and position composition through three phases Illustrative volumes. Fees follow volume; the position drifts to ASTRX as the price goes to the floor and to stablecoin as it goes to the ceiling.
A note on a corridor-aware pool
Uniswap v4 lets a pool set its fee per swap through a hook. A hook that knows the corridor could charge less near the edges, where arbitrage does the protocol’s work, and more in the middle, where traders pay for the band. This is a design idea for a later stage, not a feature of the protocol: nothing on this page assumes it exists, and it would be a pool’s choice, not the contract’s.
Where the study stops
Volumes are illustrative; the mechanism is not. Before liquidity exists the bounds are theoretical, and the founders’ initial position is what makes them real. Gas on Arbitrum can spike; the figures use ordinary conditions. A protocol fee on v4 pools reduces the provider’s take by the share the governance sets, not the trader’s cost.
Conclusions
What follows for the corridor, for the provider, and for the choice of pool.
- The band is the fee tier
The realised corridor width is set by 2φ + f_t plus slippage. The protocol publishes the bounds; the pool decides how close to them the price lives.
- Both loops are the protocol’s work, done for pay
The upper loop is the saver’s upward channel in another hand; the lower is neutral redemption in another hand. Neither harms backing; the upper raises it.
- Impermanent loss is bounded
Because the price cannot leave the range, the provider’s worst case is holding one side of the pair at an edge — and the lower edge is defended.
- The multiplier makes the floor soft on purpose
In drawdowns the lower loop’s threshold moves down with 1/(1 − drawdown), so there is no moment before which leaving pays.
- Re-ranging is profit-taking
On a corridor that rises by rule, moving the range up sells ASTRX at a higher level. The cost is gas, and gas is cents.
- Launch at 0.3 %
A tight enough band to be a corridor, a wide enough fee to pay a provider. 1 % is for the day liquidity is scarce; 0.05 % for the day it is abundant.
- Range the founders’ position on the corridor
From the effective floor to the mint price, moved up on a cadence — not a single wide range set once.
- Consider a corridor-aware hook later
Lower fees at the edges, higher in the middle. A pool’s choice; not part of the contract.
Volumes in the scenarios are illustrative. The bounds require arbitrageurs, who require liquidity. A v4 protocol fee lowers the provider’s take, not the trader’s cost.