The Fubini Null: Or, How I Used Math and First Principles to Detect Unobserved Variable Bias, Which Also Resolves Simpson’s Paradox

This paper presents a novel mathematical framework that resolves Simpson’s Para-dox by detecting unobserved variable bias through integration order dependence. Bytreating Fubini’s theorem as a null hypothesis, I develop a first-principles approachthat can determine which aggregation level in paradoxical data is valid for causal infer-ence—without requiring domain knowledge. The framework provides 2n − 1 indepen-dent tests for an n-variable system, where test failure indicates non-orthogonal hiddenvariables that bias causal relationships. When tests pass, temporal-spatial covarianceanalysis can proceed reliably. This represents the first principled, domain-agnosticmethod to resolve the aggregation level ambiguity that defines Simpson’s Paradox.The mathematics are developed entirely from first principles, making the work self-contained and accessible while providing practitioners with a concrete tool for one ofstatistics’ most persistent challenges.Keywords: Causal inference, Simpson’s paradox, Fubini theorem, unobserved variablebias, quantitative methods, mathematical psychology, computational modeling, statis-tical methodology, order dependence, domain-agnostic inference