Complexity of real-time Gaussian environments: is there really a $T_{\max}$-dependence?

The challenge in simulating Gaussian environments often lies in constructing efficient representations of the bath. In this work, we establish a framework for analyzing the complexity of Gaussian environments. We prove that the complexity is independent of $T_{\max}$ for mildly singular spectral densities, and grows only logarithmically when stronger singularities are present, such as a step discontinuity, logarithmic divergence or power law divergence. We also show that the complexity is independent of the inverse temperature $\beta$. Our results clarify the origin of the $T_{\max}$-dependence of complexity in Gaussian environments, and provides a rigorous foundation for practical simulations of open quantum systems.