Planetary Convexity: A Constraint Law of Survival Under Inevitable Pressure

Long-lived systems operating under persistent, unavoidable pressures exhibit a common structural response: dominant failure pathways are reorganized into primary gain channels. This paper formalizes that mechanism as convex inversion, a discontinuous reconfiguration in which exposure to an inevitable pressure increases system viability rather than diminishing it. The framework distinguishes structural reorganization from incremental adaptation, introduces a diagnostic criterion based on viability gradients, and characterizes convex inversion as a threshold phase transition accompanied by hysteresis. Canonical exemplars are examined across biological and planetary substrates, including gut parasitism, atmospheric oxygenation, and population variance management under environmental uncertainty. Entropy accounting and falsifiability conditions are provided. Planetary Convexity offers a unifying constraint law for biological and planetary persistence under metabolizable pressures.