A Field Equation from Relativistic Predicts Critical Transitions in Biological and Biogeochemical Systems

Abstract BACKGROUND The unification of continuum and quantum descriptions remains unresolved. Existing approaches add mathematical complexity without axiomatic simplicity. METHODS We axiomatize the vacuum as an information-energy ground state and derive a field equation from relativistic first principles. Three axioms yield five theorems, producing the master equation α = ln 2 + (1 − ln 2)(Q − C), where Q and C are quantum and entropy responses. All parameters are derived analytically; none are fitted. The predictions are tested against two independent experimental datasets extracted from the literature: plant-defense density transitions and soil-carbon stabilization kinetics. RESULTS The vacuum constant α₀ = ln 2 ≈ 0.6931 and the avalanche threshold R_c = 2/ln 2 − 1/(ln 2)² ≈ 0.8043 emerge as exact mathematical consequences. In plant-defense systems, the observed transition point R ≈ 0.81 agrees with R_c within 1%. The measured threshold contraction slope yields α = 0.70 ± 0.05, agreeing with ln 2 within 1.5%. In soil-carbon systems, manure amendment drives an avalanche transition crossing R_c between months 6 and 12, whereas biochar follows asymptotic convergence; both converge to the predicted stable state. These experimental systems reveal two distinct stabilization routes—Iron Gate and Enzyme Latch—not previously classified within a unified theoretical framework. CONCLUSIONS A single field equation, derived without empirical fitting, predicts critical transitions across physical, biological, and biogeochemical domains and yields new mechanistic classifications in experimental systems.