A pool can match a desk paying the external spread on every rebalance and still be costly to hedge. Fukasawa proves the match under continuous prices and pure arbitrage, provided either the pool fee or the external spread is positive. The comparison is an accounting identity: the automated market maker (AMM) and a desk making the same trades at the external bid and ask finish with equal liquidation wealth. Its practical use may lie in design. An LP can choose a decreasing inventory schedule or a constant weight band, then let arbitrageurs carry out the trades.
Where the equality comes from
The model holds one risky asset and a numéraire. A continuous reference price S* anchors the external bid (1−λ)S* and ask S*/(1−λ). The pool takes a fee τ on incoming assets. Fukasawa considers the Uniswap v3 type, where fees accrue in separate accounts while active reserves follow Y = f(X), and the Uniswap v2 type, where retained fees raise the level of the trading function.
Buying from the pool pays only if the arbitrageur can sell at the external bid. Selling to it requires a purchase at the external ask. Those constraints put the pool price in γS* ≤ S ≤ S*/γ, where γ = (1−τ)(1−λ). This band acts like a frictionless market with effective fee τ + λ − τλ, although the LP earns fees from τ alone. Trading stops inside it. At its edges, reflection moves reserves back into the band, leaving inventory continuous and of finite variation whenever τ + λ > 0.
For comparison, the paper constructs a self-financing external strategy holding the pool's total risky inventory Q, including fees wherever they accrue. It buys at the ask, sells at the bid and marks liquidation at the bid. Transaction-cost LVR is that strategy's wealth minus the pool's wealth. Pure arbitrage reaches only the band edges, where the pool's ask equals the external bid on a sale and its bid equals the external ask on a purchase. The cash flows therefore agree per unit: dLVR = (S^b−A)dQ↓ + (B−S^a)dQ↑ = 0 under either fee mechanism. As the paper puts it, "Thus loss-versus-rebalancing relative to this transaction-cost benchmark is zero, even without a pool fee if the external spread is positive."
The fee-free loss (1/2)∫f''(X) d⟨X⟩ requires quadratic variation in inventory. Reflected inventory has none, leaving the loss at the point Fukasawa calls singular, (τ, λ) = (0, 0). The words "relative to this transaction-cost benchmark" matter: the desk in this comparison also pays the external spread. Equation (35) gives the other side of the trade. Flow inside the band produces dLVR ≤ 0, allowing non-arbitrage takers to put the LP ahead of that desk. The zero result concerns the extra cost of arbitrage flow.
Hedging has a bill
Proposition 4.1 prices an external hedge of −Q. Combined wealth falls by (S^a − S^b) × dTV(Q), the spread multiplied by total inventory turnover. At small λ, that spread is roughly 2λS* (our arithmetic from the paper's quote definitions). Fukasawa writes, "This cost is shared by the pool and benchmark and hence cancels from their relative comparison." A delta-neutral LP desk needs to estimate the turnover in Proposition 4.1. Zero LVR leaves this hedge bill intact.
What can arbitrageurs implement?
One construction fixes an absolute inventory schedule X = q(S). It accepts any continuously differentiable q satisfying q' < 0 and ∫₀^{p₀} q(r)dr < ∞. The pool sets u = −exp{f(x) − y}; its marginal price is independent of the numéraire reserve. Retained fees accumulate as excess numéraire K = Y − f(X), with dK ≥ 0, while X = q(S) remains exact. The power law c·p^(−β), with 0 < β < 1 and zero numéraire buffer, produces a geometric-mean curve with k = (1−β)/β. A 1/p schedule across the whole positive axis is inadmissible.
The other construction aims at portfolio weights. It takes constant no-trade boundaries π− < π+ as inputs; the paper cites Davis and Norman and Shreve and Soner on their origin. Set a = π−/(1−π−), b = π+/(1−π+), weight k = √(ab), band γ = √(a/b), and pool fee τ = 1 − γ/(1−λ). With retained fees, the geometric-mean pool buys at π− and sells at π+.
The implied fee can be large. A 45% to 55% band gives a ≈ 0.818 and b ≈ 1.222 under the paper's formulas. It follows that k = 1 (constant product) and γ ≈ 0.818; with negligible λ, the required pool fee is near 18%. Widening the weight band lowers γ and raises τ. The external spread imposes a floor too: γ ≤ 1−λ, so a band tighter than the spread permits would require a rebate. Under separate fee accounts, weights drift from their targets. The paper says exact implementation there requires continuous active management.
Continuous prices are doing the work
The equality assumes continuous external prices and arbitrage exactly at the band edge. Section 6 loosens both conditions. Interior flow helps the LP, while a jump can push the pool outside the band and let arbitrageurs trade against stale quotes. The paper concludes that zero LVR does not extend directly to price jumps. My inference, which the paper does not make, is that block-time arbitrage on a real chain resembles a series of small jumps more than reflection. Gas, protocol fee deductions and deposits or withdrawals are excluded.
We could not test any of this. Our crypto price bars contain no pool reserves, swaps or fee accruals, and no executable external bid and ask. The benchmark depends on precisely those quotes.
I would put more weight on the designs if live-pool measurements of transaction-cost LVR showed a small jump component at realistic block times. As it stands, the theorem confines the quadratic-variation loss of fee-free pools to the singular point λ = 0. The fee and k formulas still tell an LP what band it is buying.