If Bonart is right, your market-flow propagator is calibrated on somebody else's trades. The paper explains how arbitrage could produce a trader-specific impact law without anyone observing her executions.
Who has to know the trader?
Bonart's earlier paper argued that prices remain diffusive whether or not a given trader participates. For information-neutral flow, that requires her impact returns to be white. The formal result is j = cU·w: w is the innovation in her flow, and U is a causal all-pass filter. U can shift timing while preserving power at every frequency. Since her flow is q = L·w, impact expressed in trades follows cUL⁻¹. Her own autocorrelation therefore enters the law. Who attributes the trades to her? We raised that gap in an earlier review.
The new paper supplies a linear-Gaussian economy with one exogenous trader, N arbitrageurs and M independent components of fundamental return. Each arbitrageur observes public returns and a private signal about one component. In the symmetric case, that signal's squared correlation with fundamental return is 1/N. Each sees little, and none sees the trader's flow. They trade under Almgren-Chriss permanent and temporary impact, repeatedly taking damped optimal positions against the predictable part of a weighted markout.
A lone Kalman-filtering arbitrageur whitens returns, Bonart shows, but leaves a problem: the trader's impact loads on past counterfactual returns. The lag coefficients are −ρk[ρ(1−k)]^(ℓ−1), and impact can no longer be expressed through her flow alone. With many arbitrageurs, their private signals jointly cover all fundamental components when M = N. Bonart calls this aggregate informational completeness. The counterfactual channel then disappears; residual impact depends on trader innovations and an initial state. There is no data anywhere in the paper. On our reading, this is pure theory.
The finite-window result
The proof works on a finite window. Shocks occupy a space of dimension T(M+1)+d0, allowing Bolzano-Weierstrass to supply convergent subsequences. At every limit point, returns are white and uncorrelated with privately observed past fundamentals. Full convergence of the arbitrage sequence remains open. Its steps are only square-summable; Bonart says absolutely summable forecasts would give a Cauchy argument.
Turning that result into an all-pass law also requires a stationary, time-invariant limit. Bonart says the limit "shall remain a technical assumption in this paper." Under that assumption, the all-pass has order at most d0, the dimension of the pre-window state. Section 3 restates the square-root law from the companion paper rather than deriving it here.
The microfoundation is genuine, and conditional.
Why should impact cost anything?
White impact can leave trading free. With pure surprise impact, U = I, expected impact cost is C[I] = 0. The paper does not prove which phase arbitrage selects, although it notes that damping can create phase. Bonart instead chooses the U with the greatest overlap ⟨U, L⟩ with the trader's flow filter. He calls the criterion minimum distortion: perturb her flow as little as possible.
For any admissible flow with L[0] > 0, every overlap maximizer obeys C[U] ≥ J[U]/2 ≥ L[0]/2. Costs thus reach at least half the flow-impact overlap. The model does not deliver the choice of phase. Bonart writes, "To be clear, this section does not establish that such a favored all-pass is attainable by the arbitrage process." The conclusion likewise names phase selection from trading behavior as a central next step. Every subsequent number describes the phase he chose.
The 50% to 60% band
The decay figure comes from a biexponential flow kernel, L[t] = (1−z1)z1^t − (1−z2)z2^t. Bonart calls it the simplest form his model allows. Its optimal phase is first-order. The cost-to-peak ratio is (2+b)/(3+z1z2+2b), where b = √((1+z1)(1+z2)). For any pair of decay rates, the ratio lies strictly between 1/2 and 3/5. It tends to 3/5 as both rates go to 0, and to 1/2 as both go to 1. Mean cumulative impact falls 50% to 60% from its peak, leaving 40% to 50%.
The abstract calls that band "lower than some empirical estimates and quite consistent with others," then cautions that "more complicated flow models can lead to different decays." Taken together, those claims are fair. The band is below the roughly 2/3 permanent impact reported by Moro et al. 2009, Zarinelli et al. 2015 and others. It is close to the about 0.42 of peak that Bucci et al. 2019 find after deconvolution.
The introduction makes a stronger comparison. It calls the prediction "very close" to known long-term decay values for heavily autocorrelated metaorder streams, citing Gomes and Waelbroeck 2015 and Bucci et al. 2019. Those streams are a narrower population than the one behind the generic permanent-impact figure. The paper does not specify which Gomes and Waelbroeck figure it means, while its later comparison places Gomes and Waelbroeck 2015 among the roughly 2/3 studies. Bucci et al. 2019 also appears both among the roughly 2/3 next-day impact studies and as the source of the 0.42 deconvolved value. Read consistently, the support comes down to one deconvolved number.
Relaxation raises another issue. For a real rational all-pass, U(1) = ±1: an isolated impulse never relaxes. Bonart argues that isolated impulses do not naturally occur, so permanent impact from trading embedded in a flow can behave well. He also shows that, for every 0 ≤ α < 1 and every L, some all-pass makes permanent impact a fraction α of contemporaneous impact. Full relaxation requires an addition. His low-pass distortion has a free parameter ρ and squared distance (1−ρ)/(1+ρ) from U, vanishing as ρ approaches 1. It brings weak long-run mean reversion, but the paper does not fit ρ.
The discussion presses its evidence too far. Bonart calls informational completeness "not immediately compelling" and offers "plausible a priori reasons to believe in it". He shows that linear feedback through the impact state preserves whiteness. He then says the framework's reproduction of known empirical impact phenomenology "provides a strong indication" that markets are close to informational completeness. Yet completeness enters the argument to produce diffusive impact, from which the phenomenology follows. Another route to diffusive impact would reproduce it equally well.
A test a desk could run
We could not backtest this. Estimating one trader's L and her innovations w requires participant-level signed executions and schedules. One-minute bars combine participants' flows and obscure the immediate response to individual executions.
Bonart identifies the discriminating experiment and says it remains outstanding. Tóth et al. 2018 found no significant difference between an asset manager's propagator and the market's, but that manager's flow statistics resembled the market's own. A desk whose fills are known and whose flow autocorrelation clearly differs from the tape's can compare its own-fill kernel with the anonymous market kernel. A match would weaken the participant-specific law's main prediction.