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arxiv: 2606.08805 · v1 · pith:AZAWY6VMnew · submitted 2026-06-07 · ❄️ cond-mat.dis-nn · q-bio.NC

Predictable Mean-Field Chaos in Random Recurrent Networks

Alkesh Yadav , Vladimir Shaidurov , Jonathan Kadmon This is my paper

Pith reviewed 2026-06-27 17:24 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn q-bio.NC
keywords mean-field theoryrandom recurrent networkschaospredictabilityKrylov subspaceLyapunov exponentdynamical systemsneural networks
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The pith

For analytic nonlinearities with fast Fourier decay, the continuous past of any realized mean-field trajectory uniquely determines its future.

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

Dynamical mean-field theory recasts deterministic chaos in random recurrent networks as an effective stochastic process. The paper shows that when the nonlinearities are analytic and their Fourier transforms decay sufficiently fast, this stochasticity is only apparent. A realized trajectory's continuous past fixes its future, so the mean-field description becomes a conditional predictor for individual paths rather than solely an ensemble average. The latent determinism is organized as an infinite hierarchy of temporal modes revealed by unfolding the power spectrum into Krylov state space. The growth rate of that hierarchy sets the complexity of finite-resolution prediction and upper-bounds the largest Lyapunov exponent, separating microscopic sensitivity from mean-field predictability.

Core claim

The continuous past of a realized mean-field trajectory uniquely determines its future. The mean-field theory is therefore not merely an ensemble description, but a conditional prediction theory for individual trajectories. Unfolding the power spectrum into a Krylov state space exposes how this latent determinism is organized across an infinite hierarchy of temporal modes. The associated Krylov growth rate sets the complexity of finite-resolution prediction and upper-bounds the largest Lyapunov exponent in this class of networks. Thus, microscopic sensitivity and predictive complexity are distinct aspects of mean-field chaos.

What carries the argument

The Krylov state space constructed by unfolding the power spectrum, which organizes the latent determinism of each mean-field trajectory into an infinite hierarchy of temporal modes.

If this is right

  • Mean-field theory supplies conditional predictions for single trajectories rather than only ensemble statistics.
  • The Krylov growth rate determines the complexity of finite-resolution prediction from the mean-field description.
  • This growth rate upper-bounds the largest Lyapunov exponent of the network.
  • Microscopic sensitivity to initial conditions and the predictive complexity of the mean-field process remain distinct.

Where Pith is reading between the lines

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

  • The deterministic structure may allow reconstruction of long-term mean-field behavior from finite past segments when resolution is limited.
  • Similar Krylov unfolding could be tested in other dissipative systems where mean-field approximations produce apparent noise.
  • The separation of sensitivity from predictability suggests that ensemble-level statistics can remain chaotic even while individual mean-field paths are conditionally deterministic.

Load-bearing premise

The nonlinearities are analytic functions whose Fourier transforms decay sufficiently fast.

What would settle it

A numerical realization of a mean-field trajectory in a network with analytic nonlinearity where two paths sharing identical continuous pasts diverge at later times.

Figures

Figures reproduced from arXiv: 2606.08805 by Alkesh Yadav, Jonathan Kadmon, Vladimir Shaidurov.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: illustrates this construction for the sech ker￾nel. As p increases, the finite predictor better approx￾imates the exact spectrum, the conditional variance re￾mains small for longer times, and the prediction hori￾zon grows. A WKB analysis (Sec. S9 of [9]) finds a largest τp ∼ log p for which σ∞(τp) remains small, show￾ing that exact predictability is recovered gradually from finite-resolution observations. … view at source ↗
read the original abstract

Dynamical mean-field theory recasts deterministic chaos in random recurrent networks as an effective stochastic process. We show that for analytic nonlinearities with sufficiently fast Fourier decay, this stochasticity is only apparent: the continuous past of a realized mean-field trajectory uniquely determines its future. The mean-field theory is therefore not merely an ensemble description, but a conditional prediction theory for individual trajectories. Unfolding the power spectrum into a Krylov state space exposes how this latent determinism is organized across an infinite hierarchy of temporal modes. The associated Krylov growth rate sets the complexity of finite-resolution prediction and upper-bounds the largest Lyapunov exponent in this class of networks. Thus, microscopic sensitivity and predictive complexity are distinct aspects of mean-field chaos. Our results extend Krylov growth ideas developed for Hamiltonian chaotic dynamics to classical dissipative systems.

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 / 3 minor

Summary. The manuscript claims that dynamical mean-field theory for random recurrent networks yields only apparent stochasticity when the nonlinearities are analytic with sufficiently fast Fourier decay. In this case the continuous past of any realized mean-field trajectory uniquely determines its future, converting the mean-field description into a conditional predictor for individual trajectories. The power spectrum is unfolded into an infinite Krylov hierarchy whose growth rate both sets the complexity of finite-resolution prediction and upper-bounds the largest Lyapunov exponent, thereby separating microscopic sensitivity from predictive complexity. The argument extends Krylov-growth techniques from Hamiltonian to classical dissipative systems.

Significance. If the central claim is established, the work supplies a concrete mechanism by which mean-field theory becomes predictive rather than merely statistical, and it cleanly separates two notions of chaos that are often conflated. The explicit conditioning on analyticity plus Fourier decay, the use of unique continuation, and the parameter-free character of the Krylov bound are genuine strengths that could influence both theoretical neuroscience and the broader study of high-dimensional dissipative chaos.

minor comments (3)
  1. [Abstract] The abstract asserts that the Krylov growth rate 'upper-bounds the largest Lyapunov exponent' but does not indicate where this inequality is proved or whether it is sharp; a brief pointer to the relevant theorem or section would help readers locate the argument.
  2. [Abstract] The phrase 'sufficiently fast Fourier decay' is left qualitative in the abstract; the manuscript should state the precise decay condition (e.g., exponential or super-exponential) required for the unique-continuation step.
  3. Notation for the mean-field trajectory and its power spectrum should be introduced once and used consistently; a short nomenclature table or inline definition list would reduce ambiguity when the Krylov unfolding is presented.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary, positive assessment of significance, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's core claim rests on the mathematical fact that analytic functions with rapid Fourier decay admit unique continuation, making the mean-field trajectory's future determined by its past. This is presented as a direct consequence of the stated assumptions rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The Krylov unfolding and growth-rate bound are derived from that unique-continuation property; no equation reduces to its own input by construction, and the extension of Krylov ideas is framed as an application to dissipative systems without circular reduction to prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption of analytic nonlinearities with fast Fourier decay; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Nonlinearities are analytic with sufficiently fast Fourier decay
    This condition is required for the determinism to hold as per the abstract.

pith-pipeline@v0.9.1-grok · 5668 in / 1072 out tokens · 16777 ms · 2026-06-27T17:24:42.564569+00:00 · methodology

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Works this paper leans on

28 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Sompolinsky, A

    H. Sompolinsky, A. Crisanti, and H. J. Sommers, Phys. Rev. Lett.61, 259 (1988)

    1988

  2. [2]

    Kadmon and H

    J. Kadmon and H. Sompolinsky, Phys. Rev. X5, 041030 (2015)

    2015

  3. [3]

    Helias and D

    M. Helias and D. Dahmen,Statistical field theory for neu- ral networks, Vol. 970 (Springer, 2020)

    2020

  4. [4]

    Engelken, A

    R. Engelken, A. Ingrosso, R. Khajeh, S. Goedeke, and L. Abbott, PLoS computational biology18, e1010590 (2022)

    2022

  5. [5]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, Phys. Rev. X9, 041017 (2019)

    2019

  6. [6]

    Krylov Complexity

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Krylov complexity (2025), review article, CERN- TH-2025-128, arXiv:2507.06286 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  7. [7]

    V. S. Viswanath and G. M¨ uller,The Recursion Method: Application to Many-Body Dynamics, 1st ed., Lecture Notes in Physics Monographs, Vol. 23 (Springer Berlin, Heidelberg, 1994) p. 262

    1994

  8. [8]

    Teschl,Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical surveys and mono- graphs (American Mathematical Society, 2000)

    G. Teschl,Jacobi Operators and Completely Integrable Nonlinear Lattices, Mathematical surveys and mono- graphs (American Mathematical Society, 2000)

    2000

  9. [9]

    See Supplemental Material at [URL] for detailed deriva- tions

  10. [10]

    R. Kubo, M. Toda, and N. Hashitsume,Statistical Physics II: Nonequilibrium Statistical Mechanics, Vol. 31 (Springer Science & Business Media, 1991)

    1991

  11. [11]

    A. M. Yaglom,An Introduction to the Theory of Station- ary Random Functions(Prentice-Hall, 1962) translated by R. A. Silverman; reprinted by Dover Publications, 1973

    1962

  12. [12]

    Dym and H

    H. Dym and H. P. McKean,Gaussian Processes, Func- tion Theory, and the Inverse Spectral Problem, Probabil- ity and Mathematical Statistics (Academic Press, New York, 1976)

    1976

  13. [13]

    R. E. A. C. Paley and N. Wiener,Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Vol. 19 (American Mathemat- ical Society, Providence, RI, 1934)

    1934

  14. [14]

    D. S. Lubinsky, H. N. Mhaskar, and E. B. Saff, Construc- tive Approximation4, 65 (1988)

    1988

  15. [15]

    Crisanti and H

    A. Crisanti and H. Sompolinsky, Phys. Rev. E98, 062120 (2018)

    2018

  16. [16]

    Engelken, F

    R. Engelken, F. Wolf, and L. F. Abbott, Phys. Rev. Re- search5, 043044 (2023)

    2023

  17. [17]

    Balucani, M

    U. Balucani, M. Howard Lee, and V. Tognetti, Physics Reports373, 409 (2003)

    2003

  18. [18]

    Haydock and C

    R. Haydock and C. M. M. Nex, Journal of Physics C: Solid State Physics18, 2235 (1985)

    1985

  19. [19]

    M¨ uck and Y

    W. M¨ uck and Y. Yang, Nuclear Physics B984, 115948 (2022)

    2022

  20. [20]

    Kadmon, Human Arenas8, 869 (2025)

    J. Kadmon, Human Arenas8, 869 (2025)

    2025

  21. [21]

    V. V. Uchaikin and V. M. Zolotarev,Chance and Stabil- ity: Stable Distributions and Their Applications(VSP, Utrecht, 1999)

    1999

  22. [22]

    Teschl,Ordinary Differential Equations and Dynami- cal Systems, Graduate studies in mathematics (American Mathematical Society, 2012)

    G. Teschl,Ordinary Differential Equations and Dynami- cal Systems, Graduate studies in mathematics (American Mathematical Society, 2012)

    2012

  23. [23]

    Simon, Advances in Mathematics137, 82 (1998)

    B. Simon, Advances in Mathematics137, 82 (1998)

    1998

  24. [24]

    J. A. Murdock,Normal forms and unfoldings for local dynamical systems(Springer, 2003)

    2003

  25. [25]

    Paz´ o, Phys

    D. Paz´ o, Phys. Rev. E110, 014201 (2024)

    2024

  26. [26]

    Schuecker, S

    J. Schuecker, S. Goedeke, and M. Helias, Phys. Rev. X 8, 041029 (2018). 7 SUPPLEMENTAL MATERIAL CONTENTS S1 Derivation of the Time-Averaged Prediction Error 7 S2 PSD Asymptotics and Imaginary-Time Singularities of Autocorrelation 8 S3 Mapping DMFT Autocorrelation Dynamics to an Effective Potential 10 S4 Analytic Structure of the Potential,V(∆) 10 S4.A Smo...

    2018

  27. [27]

    log(n) +O(1). Substituting 1−x n = ∆0 ∆0+2/k0 , we get |A2n|2 (2n)!∆2n 0 ∼K 1 ∆0+ 2 k0 2n na− 1 2 .These coefficients have the same large-nform as those of the binomial singularity  1− ∆ ∆0 + 2 k0 !2  −λ = X n≥0 (∆0 + 2 k0 )−2n Γ(n+λ) Γ(λ)Γ(n+ 1) ∆2n, Γ(n+λ) Γ(n+ 1) ∼n λ−1, withλ=a+ 1

  28. [28]

    ThusEis a sum over pairs of breakpoints

    Hence the singular part of the potential has the local form Vsing(∆)∼K 1− ∆ R −(a+ 1 2) .(S53) 17 Gaussian-smoothed piecewise-polynomialsFor a Gaussian-smoothed piecewise-polynomial function, the leading Fourier tail is more generally a finite oscillatory sum, bΦ(k) =e −k2/k0 X j cjkaj e−ikxj ,(S54) where the pointsx j are the breakpoints and the powersa ...