Scale-Independent Variation Rates of Phanerozoic Environmental Variables and Implications for Earth’s Sustainability and Habitability

The atmospheric \(\textrm{CO}_{2}\) level, global average temperature, and sea level, which are three key metrics characterizing Earth’s surface environments, underwent a series of significant changes over geologic time. Here, I investigate the variation rates of these three variables during the Phanerozoic Eon and show that they systematically exhibit scale-independent behaviors. I then derive a general mathematical form of these scale-independent patterns based on geosystem-specific assumptions and basic physical principles. From the perspective of statistical mechanics, these scale-independent behaviors appearing in the planetary-scale geological system imply that the internal dynamics and interactions of different components in the Earth system have significantly influenced its evolution and stability, which sheds light on Earth’s sustainability and habitability.