If your execution cost model applies a concave power law to the trading rate and then convolves with power-law decay t^{-gamma}, 0 < gamma < 1, that model admits a finite round trip with negative cost. At every exponent in that range. The paper proves it for f_delta(x) = sgn(x)|x|^delta with delta > 0, which is the family everyone calibrates.
Lee's claim is clean, and the mechanism is new.
What the machine actually computes
The standard transient impact model has two pieces: an instantaneous law that turns trading speed into price pressure, and a memory kernel that decays that pressure over time. Whether the two can coexist without manipulation has been sitting open. Schneider and Lillo (2019) recorded the consistency of power-law decay with a nonlinear impact function as an open problem; Abi Jaber et al. (2025) call the compatibility of square-root impact with power-law decay long-standing.
Call the first arrangement rate-inside. You take the trading rate v(s), push it through f, and convolve the result with the kernel: D(t) = integral of H(t,s) f(v(s)) ds. Cost is the trading rate times the price displacement it walks into. This is Gatheral's 2010 setup.
Lee's Theorem 3.2 says that for any nonzero kernel integrable on the Volterra triangle, nonnegative cost on every finite piecewise-constant round trip forces f to be affine. For a nonzero convolution kernel the intercept dies too, so f is linear. Corollary 3.6 puts the number on it: with H(t,s) = (t-s)^{-gamma}, 0 < gamma < 1, and f_delta(x) = sgn(x)|x|^delta, there is no manipulation if and only if delta = 1. The safe set collapses to a single line.
The mechanism is what separates this from the earlier rigidity results. Huberman and Stanzl for permanent impact and Gatheral's Lemma 4.1 for exponential decay are continuous-time arguments that compress a two-block round trip into a horizon where the kernel is effectively permanent. They need G(0+) finite. Hey, Neuman and Tuschmann (2025b, Theorem 2.4) work differently: in discrete time, by perturbing a three-trade configuration, for kernels continuous at zero, and their result is all-input rather than round-trip, with a witness carrying nonzero total volume. Power-law decay is unbounded at zero and scale invariant, so neither door opens.
What Lee does instead: pick two rates x < 0 < y with duty weights that give exactly zero mean rate but nonzero mean impact. Repeat that pair rapidly on a source interval. The pump carries no volume and, in the fast limit, no self-cost, because the kernel is integrable and the target-rate factor averages to zero. It does leave a nonzero impact moment sitting on the interval. Then you place two thin ordinary trades, one before and one after, and the later one reads the residue. The pump has zero volume and the readout is thin, and the total cost is negative. Lemma 3.7 supplies the algebra, stated for a function g with g(0) = 0: the two-rate mean impact is nonzero for some pair unless g is proportional to the identity. Theorem 3.2 applies it to g = f - f(0).
Do two-block checks certify anything?
No, and the failure is sharp. Gatheral's slow-rate condition delta + gamma >= 1 comes from a two-block analysis, and Theorem 9.7 confirms it is sharp on the sublinear side: two blocks manipulate exactly when delta < 1 - gamma. On the superlinear side there is a separate threshold delta_+(gamma), which becomes infinite exactly when gamma >= 2 - log_2(3), about 0.415.
So for gamma = 1/2 and delta = 2, no two-block round trip has negative cost. None. Yet Corollary 3.6 guarantees a finite manipulation exists. The paper is explicit that checking two-rate schedules or scanning a coarse grid does not establish safety.
How many blocks does it take at the exponents people actually quote? Table 2 gives certified witnesses. At (delta, gamma) = (1/2, 1/2), a comb with M = 64 spikes and 129 blocks has cost bounded above by -0.10963. At (3/5, 1/2), 513 blocks give -0.17409. At (1/2, 3/5), 513 blocks give -0.044262. All endpoints and rates rational, evaluated in directed interval arithmetic at forty digits, enclosure widths below 1e-30. The pattern is slow selling at unit rate punctuated by brief intense buying bursts, which is the shape Curato, Gatheral and Lillo observed in their numerical optima. Separately, (0.55, 0.45) is the pair where they found negative execution costs numerically.
The calibrated pairs from the literature (gamma around 0.4 from Bouchaud et al., delta around 1/2 from the square-root law) all sit in the manipulable set. So does everything else off delta = 1.
Moving the nonlinearity downstream
The alternative architecture aggregates signed flow through the kernel first, D = G * v, then applies a monotone readout h to the accumulated state. Theorem 4.1: nonnegative cost for every input and every continuous nondecreasing h with h(0) = 0 is equivalent to complete positivity of G, meaning both Volterra resolvents r_a and s_a are nonnegative for every a > 0. The converse is constructive, built from hinge probes, and a single smooth dead-zone readout already detects any failure.
Every completely monotone kernel is completely positive. Power-law decay is completely monotone. So square-root impact applied to the impact state after t^{-gamma} memory is safe for every input and every monotone readout (Corollary 6.3). Same two shapes, opposite verdict, purely because of composition order.
This is the result worth having. Lee also shows linear passivity is strictly weaker than the nonlinear condition: H(t) = 2e^{-t} - e^{-2t} has strictly positive spectral density 6/((1+w^2)(4+w^2)) yet g + H fails universal round-trip safety for every real g. Checking the identity-readout quadratic form is not enough once the readout can be any monotone function.
One caution, and it does not touch the power-law case. Theorem 6.2 says every real permanent shift of a completely monotone transient is round-trip safe for every readout, including shifts under which the kernel changes sign or tends to a negative level. So t^{-gamma} plus any real constant stays safe. Completely positive tails that are not completely monotone are the exposed class: Proposition 6.13 constructs such an H where, with g = -1/2, the four-block round trip (-1, -4, 4, 1) on equal half-unit cells costs below -11/1000 for one analytic strictly increasing readout.
Friction
The repair criterion is Theorem 9.5. Under a common rate cap |v| <= B, a power penalty with exponents p_j repairs the rate-inside model with a finite coefficient exactly when p_min <= 1 + delta. Without a cap you need the exponents to bracket 1 + delta.
A proportional spread is p = 1. Since 1 + delta > 1, a spread repairs the model inside a cap, with a coefficient growing like ||G||_1 B^delta, and never repairs it uncapped. Scale a manipulation by a and its negative cost grows as a^{1+delta} while the spread grows as a. Lee's own phrasing: a spread "converts manipulation into a size threshold rather than removing it." A quadratic cost alone (p = 2) fails at the other end, because a small-amplitude chattering manipulation costs -c a^{1+delta} against an O(a^2) penalty whenever delta < 1. Spread plus quadratic together repair every concave law 0 < delta <= 1. The frictionless axiom is doing all the work here, and the paper says so.
Execution constraints also restore safety. Proposition E.2 fixes a maximum block count N, a minimum dwell tau, a horizon and gamma in (0,1), and gives a quantitative safe neighborhood of delta = 1. The witnesses in Table 2 use 129 to 513 blocks on a normalized horizon, with unbounded rate ratios in the chattering limit. No minimum dwell is imposed anywhere in the main theorems.
For permanent memory the paper computes the exact critical cubic friction: kappa_* = 0.0881331300961391335081086746213. The extremizer is a smooth one-hump inventory carrying a single rate sign change. At T = B = 1, delta = 2 and a cubic penalty, two blocks reach only 0.0750707765001944, and an exact rational eighty-block certificate reaches 0.0881070201184047938.
What this is worth
We could not test any of it. Calibrating or falsifying these architectures needs signed order flow and trade-level prints to separate the instantaneous law from the decay kernel; we have 1-minute OHLCV bars with no trade direction. Simulating the theorems on assumed kernels would only check arithmetic.
Take it for what it is: a no-dynamic-arbitrage screen on a specified frictionless model. It says nothing about whether observed impact is concave (it is) or whether the rate-inside model fits data well. The witnesses the paper displays use 129 to 513 blocks; the only proved lower bound is three, in the region 1 - gamma <= delta <= delta_+(gamma) with delta not 1.
What it does say is that the rate-inside model with concave impact cannot be defended as manipulation-free by tuning exponents, and that a two-block sanity check will hand you a false pass. Choosing where to put concavity in a cost model that carries transient decay is now a one-sided choice. The state-outside form has a clean sufficiency theorem behind it. The rate-inside form has a rigidity theorem against it.
Lee notes that the proofs were developed with substantive generative-AI assistance. Appendix F states that all universal implications are proved analytically, and the supplement checks specified formulas and witnesses. So the numeric certificates are machine-checkable and the proof chain is not machine-checked, which is a real cost of adoption for anyone building on Theorem 3.2 rather than just Corollary 3.6.
A dwell-constrained manipulation at (delta, gamma) around (0.5, 0.5) would change how I read this. The paper does not construct one, and it does not evaluate the dwell-and-block-constrained problem away from delta = 1. I did not find one either.