2026/03/13 by Bernd von Mallinckrodt · 1 voice
paper · doi:10.5281/zenodo.19002281
openalex publication_date 2026/03/13 · openalex created_date 2026/03/14 · openalex updated_date 2026/07/01
This theoretical addendum provides a rigorous mathematical analysis of the CRTI observable T = Tr(Σ)/Tr(Σ⁻¹) introduced in the companion preprint (von Mallinckrodt, 2025). We demonstrate that T is a scalar invariant of the Fisher information metric combining total fluctuation power (static susceptibility) and total system stiffness. Near stability boundaries, T diverges quadratically faster than variance alone due to a double amplification mechanism: the soft-mode eigenvalue simultaneously inflates the numerator and suppresses the denominator. Key results:• Tr(Σ) is identified as the total static susceptibility via the fluctuation–dissipation theorem• Tr(Σ⁻¹) corresponds to total system stiffness (restoring force)• T is a basis-invariant scalar invariant of the Fisher metric tensor• The divergence is universal across all stochastic systems admitting linearisation near a stable fixed point The derivation requires only three ingredients — linearised dynamics, stochastic forcing, and a Lyapunov covariance — making CRTI applicable to ecological, neural, genetic, and macroeconomic systems. Fisher information · susceptibility · soft mode · critical transitions · fluctuation-dissipation theorem · information geometry · Lyapunov equation · early warning signals · statistical mechanics · complex systems · instability · covariance matrix