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 T ₖ₈₍∝Twin) and envelope-induced suppression of the forward-model sensitivity (C (Twin) C (T ₖ₈₍) 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) T ₖ₈₍^* = Tc/ (2 - 1) Twin∗=Tc/ (2γ−1). The framework provides a quantitative criterion for when dynamic spectral information is accessible.
Hiroyuki Shioiri (Mon,) studied this question.