The plain differential model captures nearly all the active-band forecast gain on ARB5 and ETH5. These are the two 5bp WETH-USDC pools the authors judge compatible with their stationary specification. Both show enough mean reversion for every fitted model's 8-hour forecast to beat "no change" on the active band. The authors describe the error ratios across configurations as "nearly common". Their distinctive addition, a nonlocal relocation operator, has yet to earn its parameters on this measure. The empirical result is limited, and the study stops short of LP P&L.

From liquidity ranges to a curve

A Uniswap V3 LP supplies liquidity within a price range. Add up the overlapping ranges and the result is a liquidity profile: the depth available at each price. Lee, Risk, Tung and Wang use the logarithm of that profile, centered on spot, as their state variable. The whole curve enters the model.

Low-order polynomials separate its movements into modes. Mode 0 measures level, or total depth. Mode 1 measures slope, the tilt of depth to either side of spot. Mode 2 measures curvature, or how closely depth clusters around spot.

Two operators govern the drift. A degenerate Sturm-Liouville operator damps mode n at rate a·n(n+2α), making higher modes revert faster. The concentration order α selects the polynomial family. A Hilbert-Schmidt relocation operator V then allows movements in one mode to feed another, representing LPs shifting entire ranges at once. The differential part gives mode 0 zero damping. As the authors put it, "the differential operator supplies no damping of the level mode". Any mean reversion in the level must therefore come through V.

The theory establishes a unique mild solution and expresses the model as coupled linear SDEs. It also supplies moment formulas for reserves, pool value and loss-versus-rebalancing (LVR).

The sample contains three WETH-USDC pools: 5bp on Arbitrum (ARB5), 5bp on Mainnet (ETH5) and 30bp on Mainnet (ETH30). It runs from August 2023 to early October 2024. Roughly 8 hours separate snapshots, yielding 1,200 observations divided into 400-observation windows. The reported results cover only W1 and the Full block. Within each window, estimation uses the first half; forecasts then roll one step ahead through the second half.

The authors fit four configurations. V=0 leaves the level as a random walk. A sparse V acts on the level and slope diagonals, while a dense 3x3 V allows more interactions. The remaining configuration applies a Frobenius penalty of 3.0 to the dense V.

R_A, the headline measure, divides forecast RMSE by the martingale's RMSE over the central half of the grid. A ratio below 1 wins. This measures a forecast of the depth curve; earning fees or reducing LVR would require separate evidence.

Does V earn its place?

On ARB5 and ETH5 active-band error, the answer is mostly no. ARB5 Full records R_A of 0.931 for V=0, 0.926 for sparse, 0.948 for dense and 0.929 for penalized. ETH5 Full records 0.843 for V=0, 0.842 for sparse and 0.854 for dense. Leaving V out already produces the 7% gain over the martingale on ARB5 Full and the 16% gain on ETH5 Full.

The authors acknowledge how similar the active-band ratios are across configurations. They argue that V remains necessary for the level: without it, that mode follows a random walk. Their reading of the data favors a sparse, bounded V over removing it altogether. They also write that "the fit leans toward the Legendre case with sparse bounded relocations". They interpret this as a regularized outcome and decline to take it as evidence that α is 1/2. Why the conclusion nevertheless credits V with "jointly interpretable estimates" remains open.

The strongest case for V comes from the level half-life. On ARB5 W1, a sparse V̂00 of 0.088 per step implies about 7.9 steps, roughly 2.8 days given the realized 8.6 hours per step. This is comparable to the roughly 3-day level persistence in Risk, Tung and Wang's 2025 study. A one-step test sees little of that effect. At 0.088 per step, damping closes about 8% of the distance to the mean during one 8-hour step (exp(-0.088) is 0.916).

The value-weighted score R_V does register a difference. It weights the profile using the pool-value formula. On ARB5 Full, sparse V scores 0.780 versus 0.864 for V=0. The unpenalized dense fit scores R_V 1.150 on ARB5 Full, consistent with the authors' description of it as over-parameterized; the penalty lowers the score to 0.807.

The PCA-based Factor VAR benchmark has higher out-of-sample severity than every operator fit on ARB5 and ETH5. Its ARB5 Full results are R_A 0.966, R_V 1.214 and out-of-sample whiteness severity 2.879.

The data pin down α weakly. For the unpenalized dense fit on ARB5 and ETH5, the five best grid values occupy a band of width 0.2. The objective changes by less than 3×10^-3 across that band. α̂ spans 0.35 to 0.80 on ARB5 and 0.40 to 1.25 on ETH5. The authors disclose these results.

Failure at the 30bp Mainnet tier

The pair stays the same, but forecasts fail at the 30bp tier. Across both blocks and all four operator fits, R_A ranges from 1.322 to 1.866. Full-block R_V reaches 2.527 to 3.079, compared with 1.043 to 1.310 on W1.

Hosking severity measures residual whiteness, with scaling that puts white residuals near 1. On ETH30 Full, it rises out of sample to 15.065 to 17.255 across the four operator fits. In sample, the range is 1.200 to 1.666. The V=0 and sparse fits offered no advance warning through in-sample whiteness: their ETH30 Full readings were 1.200 and 1.203, versus 1.171 and 1.166 on ARB5 Full. The dense fits did signal trouble, at 1.666 against 1.190.

The abstract acknowledges nonstationarity. The authors defend the framework while describing its stationary branch as strained, and they point to the fitted rate â_Δ as a warning. That defence holds in part. Every ETH30 fit has â_Δ of at most 0.005 per step, compared with 0.013 to 0.018 on the ARB5 and ETH5 Full blocks. The spectral radius, however, does little to distinguish the pools: ETH30 Full sparse has 0.998, versus 0.993 for ETH5 Full sparse.

One failure among three hand-picked pools for a single pair gives nobody a base rate they can size from.

The distance to LP P&L

The authors identify the gaps themselves:

The pool-value and LVR moments consequently never meet the data.

Independent validation would require two inputs. First, the same tick-level snapshots would need recalibration in log-moneyness coordinates instead of rank. Second, position-level mints, burns, swaps and fee accruals would allow profile forecasts to be scored against realized LP returns and LVR.

We could not perform that test. We have neither historical V3 tick distributions nor position or fee histories. OHLCV bars cannot reconstruct a liquidity surface.

A multi-horizon forecast in price-distance coordinates would change my mind about V if its 2.8-day level half-life reduced error that the V=0 random walk could not.