Contraction versus Recurrence

An Exponential Separation in Observation-Based Prediction of Deterministic Dynamics

Pavel Popovich (PPRFNK Tech, St. Petersburg)

📄 Paper: arXiv:2607.14885

TL;DR

Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state:

• Recurrence-based methods (method of analogues and descendants) wait for the trajectory to return to an ε-neighborhood of a previously observed state. Expected cost scales as ε^(−d) (Kac lemma + quantitative Poincaré recurrence), where d is the pointwise dimension of the invariant measure.
• Observer-based methods fit a converging state estimator on the delay reconstruction. A detectable linear observer converges in Θ(log(1/ε) / (1 − ρ(A_cl)²)) steps, where ρ(A_cl) is the closed-loop spectral radius of the Riccati fixed point.

Both scaling laws are verified numerically on the Lorenz attractor: return-time exponent −1.8 vs. theoretical −2.05; observer cost linear in log(1/ε) with R² = 1.000 and in (1 − ρ²)^(−1) with R² = 0.985. The measured cost gap is ~10⁹ at ε = 10⁻⁶ for d ≈ 2.

The paper also introduces an admission protocol (the "Kac–Riccati gate") that decides whether a given signal lies inside the theorem's class using surrogate-data prediction gating — and explains the recurring folklore of "universal" fractal dimensions as a dataset-size artifact bounded by 2·log₁₀N. On real data the gate admits the Santa Fe laser benchmark (D̂₂ = 2.0) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims.

All quantitative claims re-derive from a single verification script (17/17 checks).

Repository contents

File	Description
worked_example.py	Prints every number of the worked example (Sec. 1.1): the Lorenz state, the observable, the delay vectors, analog waiting times, Riccati N_a, and the 1.9·10¹⁰ gap
demon_observer_verify.py	Re-derives every number in the paper as a pass/fail assertion (17/17 at submission)
make_figs.py	Regenerates all four figures, re-measuring the data
realdata_gate.py	The E5 real-data experiment: fetches the Santa Fe laser benchmark and the SIDC sunspot series, runs the gate (laser: ADMIT, D₂ = 2.01; sunspots: REFUSE)
figures/	The four paper figures (PDF)
paper/main.tex	LaTeX source of the paper

Reproducing

Full reproduction takes minutes on a laptop.

pip install -r requirements.txt

python worked_example.py          # the Sec. 1.1 worked example, number by number
python demon_observer_verify.py   # 17/17 verification suite
python make_figs.py               # regenerates figures/fig1..fig4
python realdata_gate.py           # E5: Santa Fe laser (ADMIT) vs sunspots (REFUSE); needs network + reservoirpy

realdata_gate.py additionally requires pip install reservoirpy (Santa Fe laser data) and network access (SIDC sunspot CSV).

Citation

@article{popovich2026contraction,
  title   = {Contraction versus Recurrence: An Exponential Separation in
             Observation-Based Prediction of Deterministic Dynamics},
  author  = {Popovich, Pavel},
  journal = {arXiv preprint arXiv:2607.14885},
  year    = {2026},
  doi     = {10.48550/arXiv.2607.14885}
}

License

MIT — see LICENSE.