The Principle of Planetary Convexity

This paper proposes that Earth’s biosphere functions as a self-convex system in which all planetary variables fluctuate in service of preserving life as the invariant. From the Great Oxygenation Event to the present, catastrophe and evolution act as complementary feedbacks that stabilize the biosphere. The work develops key sub-principles — the Recursive Survival Law, Habitability Bias, and Convexity of Life — to show how Earth’s system converts instability into durability. It further introduces the conceptual equation dL/dt = f(V₁ … Vₙ) → 0 as t → ∞, expressing life’s approach to steady-state through oscillatory balance. The paper concludes with the Law of the Living Planet: Earth’s bias toward life is not chance but structure — a local law of convex stability that defines biospheric self-preservation and sets the context for cosmic humility.