Stability Is Sustainability: A Mathematical Framework for Climate Regulation, Long- Term Viability, and Legal Sufficiency

Climate regulation is often evaluated in ethical, political, or economic terms, while its physical feasibility is taken for granted. This article advances a different claim: climate sustainability is a mathematically constrained property of the Earth system, and climate regulation succeeds or fails according to whether it satisfies explicit dynamical inequalities. Drawing on the Poisson-Hamiltonian framework developed in Doucette (2025), originally formulated to explain the long-term persistence of dust clouds near the Lagrange points L4 and L5, this article recasts climate change as a stability problem for an open system subject to persistent forcing. Invariant structure, suppression of cascading feedbacks, slow-variable drift, and the balance between dissipation and replenishment together determine whether a stable climate regime can persist. A Stability-Sustainability Theorem is presented here, its inequalities are translated into climate-regulatory design principles-caps, buffers, reserves, and replenishment rates-and a judicial-style reasoning section explains why certain climate policies are mathematically insufficient regardless of political intent.