What question did this study set out to answer?

The main aim is to connect dynamical independence with the causal Fisher metric in Gaussian vector autoregressive processes.

March 4, 2026Open Access

Dynamical Independence as Causal Projection: A Fisher-Geometric Bridge with Topological Shielding at Criticality

Key Points

The main aim is to connect dynamical independence with the causal Fisher metric in Gaussian vector autoregressive processes.
Developed a theoretical framework linking dynamical independence and causal Fisher metric.
Proved fundamental relationships using mathematical formulations.
Conducted numerical verification across 54 parameter combinations with 50 random networks.
Dynamical independence correlates with the squared Frobenius norm of causal leakage.
DI-optimal projections effectively minimize causal leakage.
At criticality, macroscopic autonomy shows saturation while microscopic complexity diverges.

Abstract

We establish a formal bridge between dynamical independence (DI) and the causal Fisher metric for Gaussian VAR(1) processes, proving that dynamical dependence equals the squared Frobenius norm of the causal leakage operator weighted by the conditional Fisher metric. Three corollaries follow: DI-optimal projections minimize causal leakage ("Causal PCA"), the geometric-precision index decomposes into DI-mode contributions, and at criticality macroscopic autonomy saturates while microscopic complexity diverges (topological shielding). Numerical verification on 54 parameter combinations and n=50 random networks confirms all predictions.

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