The 3+1 Structure of Goodhart's Law: From Metric Collapse to Invisible Value Destruction

Goodhart's Law—"When a measure becomes a target, it ceases to be a good measure"—describes apervasive failure of proxy-based optimization across economics, public policy, and AI alignment.Existing formalizations either classify failure modes qualitatively (Manheim & Garrabrant, 2018) oranalyze tail distributions in static settings (El-Mhamdi & Hoang, 2024). This paper introduces adynamical formalization using the Landau-Stuart equation, treating the amplitude of proxy-goaldivergence as an order parameter undergoing a phase transition driven by optimization pressure. Thelinear growth rate σ(α) captures the onset of metric collapse: when optimization pressure α exceedsa critical threshold αc, the zero-divergence equilibrium becomes unstable and proxy-goal divergencegrows to a finite amplitude. We show that three of the four Manheim-Garrabrantcategories—Regressional, Extremal, and Adversarial Goodhart—correspond to distinct mechanismsdriving σ positive, while Causal Goodhart is structurally distinct, requiring Pearl's do-calculus ratherthan dynamical analysis. This 3+1 decomposition reflects two fundamentally different failuremodes: distributional failure under selection (amenable to Landau-Stuart dynamics) versusstructural failure under intervention (requiring causal inference). The framework yields acomputable Goodhart Number, connects to the theory of Invisible Value Destruction (Sophia,2026a), and provides concrete diagnostic criteria for AI alignment and institutional design.