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arxiv: 2606.07059 · v1 · pith:FSEOPDSZnew · submitted 2026-06-05 · 💱 q-fin.TR

Diffusive in plain sight: An inconspicuous law of market impact

Julius F. Bonart This is my paper

Pith reviewed 2026-06-27 20:18 UTC · model grok-4.3

classification 💱 q-fin.TR
keywords market impactsquare-root lawdiffusive processescounterfactual returnsinformational couplingpropagator models
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The pith

Requiring both realized and counterfactual returns to be diffusive restricts impact scaling to the square-root law when information is neutral.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines market impact as the difference between the realized return after a trade and the counterfactual return that would have occurred without the trade. By requiring both returns to follow diffusive processes, it derives an identity limiting how impact can scale with trade size at the level of each participant. This identity recovers the square-root law in the information-neutral regime. When trades carry strong information the scaling crosses over to linear. The same requirement implies that cumulative impact itself must remain diffusive in the weak-coupling limit.

Core claim

Decomposing impact as the difference between realized and counterfactual returns and requiring both to be diffusive yields an identity that restricts admissible impact scaling at the level of individual participants. This constraint implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive -- a diagnostic that many propagator and latent liquidity models fail to satisfy.

What carries the argument

The identity obtained by subtracting a diffusive counterfactual return from the realized return while preserving the diffusive property of the difference, which constrains admissible impact scaling.

If this is right

  • The square-root law holds in the information-neutral regime.
  • Impact scaling becomes linear under strong informational coupling.
  • Cumulative market impact remains diffusive in the weak-coupling regime.
  • Many propagator and latent liquidity models violate the requirement of diffusive cumulative impact.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be tested by separating trades according to their information content and measuring the resulting impact exponents.
  • It provides a first-principles reason why the square-root law appears across many empirical studies of market impact.
  • Similar diffusive constraints might be applied to other market quantities such as volatility or order flow.

Load-bearing premise

Both realized and counterfactual returns are diffusive processes.

What would settle it

Empirical data showing that the difference between realized and counterfactual returns is not diffusive, or that observed impact scaling violates the regimes predicted by the identity.

read the original abstract

Decomposing impact as the difference between realized and counterfactual returns and requiring both to be diffusive yields an identity that restricts admissible impact scaling at the level of individual participants. This constraint implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive -- a diagnostic that many propagator and latent liquidity models fail to satisfy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that decomposing market impact as the difference between realized and counterfactual returns, and requiring both to be diffusive processes, yields an identity restricting admissible impact scaling at the individual participant level. This identity implies the square-root law in the information-neutral regime and a crossover to linear impact under strong informational coupling, consistent with empirical observations. In the weak-coupling regime, cumulative market impact is itself diffusive, a diagnostic that many propagator and latent liquidity models fail to satisfy.

Significance. If the derivation holds, the work supplies a parameter-free identity linking diffusive return assumptions directly to observed impact scalings, without auxiliary fitting or simulation. Credit is due for deriving the square-root law and linear crossover as special cases of the same identity and for identifying the diffusiveness diagnostic for existing models. The approach is internally self-contained and offers a minimal-assumption route to the scaling laws.

minor comments (2)
  1. [Abstract] Abstract: the claim that the diffusive assumption 'yields an identity' is stated without any equation or derivation outline; while the full text contains the steps, a single displayed relation would improve immediate readability of the central result.
  2. [Introduction or §4] The consistency with empirical observations is asserted but would benefit from explicit citation of the specific datasets or studies invoked, particularly for the crossover regime.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, accurate summary of the manuscript, and positive assessment. The recommendation for minor revision is noted; however, no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected; derivation is self-contained from stated assumptions

full rationale

The central derivation begins from an explicit modeling choice (decompose impact as realized minus counterfactual returns) plus the assumption that both return processes are diffusive, then algebraically obtains an identity that constrains admissible impact kernels. The square-root law is shown to satisfy this identity in the information-neutral regime and is therefore a derived consequence rather than an input; the linear crossover appears under a different regime of the same identity. No fitted parameters are relabeled as predictions, no self-citations supply load-bearing uniqueness theorems, and no ansatz is smuggled in. The argument is therefore a direct mathematical implication of the diffusiveness premise and remains internally consistent without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on a single domain assumption about the statistical character of returns; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Both realized and counterfactual returns are diffusive processes.
    Explicitly stated in the abstract as the premise that yields the identity restricting impact scaling.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Empirical Confirmation of the Square-Root Law of Market Impact in a U.S. Large-Cap Equity

    q-fin.TR 2026-06 unverdicted novelty 5.0

    Empirical test on AAPL using reconstructed metaorders from full Nasdaq ITCH feed finds the square-root impact law with prefactor c_raw=0.69 consistent with global observations and preferred over linear or log forms by...

Reference graph

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32 extracted references · 4 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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