Dimensionless Reciprocal-Time Indicator for Boundary-Approach Risk in Engineered Nonlinear Systems: Theory, Falsifiable Acceptance Band, and Validation on Power-Grid and Financial Time Series

Many engineered and physical systems—power grids, mechanical structures, controlled chemical reactors, laser amplifiers, financial markets, and physiological regulators—operate close to nonlinear instability thresholds whose crossing would, in idealised limits, produce finite-time singularities in a relevant state norm. Persistent systems remain observable precisely because evolved or engineered saturation keeps them below this catastrophic regime, but the dimensionless distance from the saturation boundary is itself a structural risk indicator that is rarely measured directly. This paper develops such a measurement from first principles.

Let f be a locally Lipschitz vector field on the autonomous reduction of an engineered nonlinear system, and assume that along a candidate approach direction f is asymptotically homogeneous of degree α along the blow-up direction with leading coefficient κ > 0. We prove that the dimensionless reciprocal-time invariant Ψ(r) := Φ(t)/[r · E(t)], built from the radial energy E(t) = ½‖x(t)‖² and the radial flux Φ(t) = ⟨x(t), f(x(t))⟩ with r = 1/(T* − t), converges to a vector-field-intrinsic limit Ψ*(α) = 2 / (α − 1) for α > 1, as r → ∞ (Theorem 1). The map α ↦ Ψ*(α) supplies a one-dimensional reference manifold against which any candidate engineered system can be compared, and the gap |α − 1| furnishes a dimensionless safety margin. Under a strengthened homogeneity hypothesis controlling the next non-resonant term of the normal form, the convergence rate is sharp: |Ψ(r) − Ψ*(α)| ≤ C r^(−β) with β = δ/(α − 1) (Theorem 2).

The paper develops the indicator into an operationally complete diagnostic. We construct a quantitative reciprocal-time Lyapunov functional V(r) with explicit decay rate |V(r) − C_α| ≤ D₁ r^(−β) and singleton ω-limit set on the self-similar manifold (Theorem 3); a closed-form falsifiability inequality together with a stability theorem under finite-sample OLS estimation error in α, yielding an explicit composite acceptance band decomposed into intrinsic rate, estimator bias, and estimator variance (Theorems 4 and 5); and an additive-noise robustness theorem (Theorem 6) bounding the deviation of the empirical Ψ-estimate under observation noise and quantifying the minimum signal-to-noise ratio required for the falsifier to be admissible.

Numerical validation on two-dimensional perturbed homogeneous flows confirms convergence to Ψ*(α) within 0.2%–4.5% across α ∈ {1.5, 2.0, 3.0}, with empirical errors quantitatively identified as finite-r samples of the rate r^(−β) and shown to lie inside the derived acceptance band. Two real-world engineered/economic datasets from disjoint domains are then processed through the same pipeline as boundary-approach measurements: the Continental European synchronous power-grid frequency log of January 2019 yields α_est = 0.36 ± 0.04, and the NASDAQ Composite dot-com bubble (1994–2000) yields α_est = 0.49 ± 0.06. Both lie deep in the saturated regime α ≪ 1 as expected for persistent systems, and their non-convergent Ψ(r) profiles are correctly predicted by the framework. The relative ordering of the two systems with respect to the boundary α = 1 is resolved to better than the combined error bar, demonst