Theorems for Inferring Canonical Environmental Signature from Causal Response

We present a model-free framework for inferring environmental persistence signatures from a single causal response signal R(t). The approach is based on seven estimators of persistence timescales centroid, step-response, autocorrelation, envelope, energy, and spectral-pole timescales whose agreements and disagreements encode diagnostic information about underlying system dynamics. Four theorems establish necessary and sufficient conditions for canonical regimes: (1) τ (s) = τ (2) for step-type responses with finite lag area, (2) τ (s) = τ pole for minimum-phase LTI systems, (3) The diagnostic parameter ρ 13 = τ (s) /τ (3) = 1 if and only if memory is exponential, and (4) The diagnostic parameter ∆ su → −1 and ∆ env 23 → 0.5 if and only if the system is a linear underdamped oscillator. We validate the framework on canonical systems including RC circuits, critically damped and underdamped mechanical systems, and pharmacokinetics. The capacity of these signatures under realistic measurement noise was analyzed. The framework enables passive, excitation-agnostic classification of dynamical regimes monotonic vs. oscillatory, Markovian vs. non-Markovian, linear vs. nonlinear damping from R(t) alone, without parametric assumptions or excitation knowledge. We made available all implementation codes (open-source) in the name QSignature library on Github.