Observability and Fragility in Dynamic Noise Spectroscopy

We develop a minimal framework for dynamic noise spectroscopy in time-dependent environments, where observables probe window-averaged rather than instantaneous spectra. Using a windowed filter-function formulation and Fisher-based identifiability, we show that observability is confined to a finite band in the joint space of time and measurement window. This band arises from a scale competition between Fisher amplification (∝Twin∝ √{Twin}∝Twin) and envelope-induced suppression of the forward-model sensitivity (C(Twin)C(Twin)C(Twin)). The resulting “observable horizon” is not a geometric rank effect but a consequence of this competition. A minimal two-parameter model demonstrates the effect, and numerical results confirm the analytic scaling of the optimal window Twin∗=Tc/(2γ−1)Twin^* = T_c/(2γ - 1)Twin∗=Tc/(2γ−1). The framework provides a quantitative criterion for when dynamic spectral information is accessible.