Foundation Depth in Coordination Systems: Physical Constraints, Mathematical Structures, and Game-Theoretic Security

This paper develops a formal framework for analyzing coordination systems according to foundation depth—an ordinal classification measuring how deeply a system's essential properties depend on principles resistant to variation. We distinguish five trust layers: Physical (Φ), Mathematical (Μ), Protocol (Π), Institutional (Ι), and Social (Σ). We establish three impossibility theorems: (1) Bootstrap Impossibility (no system can ground its own initialization in natural law), (2) Semantic Gap (mappings from physical states to coordination meanings are not derivable from physics), and (3) Coordination Security (security against strategic adversaries is necessarily game-theoretic, not physical). The paper clarifies a common misconception: Bitcoin's security derives from a Nash equilibrium under specified assumptions, not from thermodynamics. Proof-of-work creates physical costs that structure a game in which honest behavior is incentive-compatible; physics provides the cost structure, not the security itself. We apply this framework to classify systems across biological, technical, and social domains, revealing systematic trade-offs between foundation depth and adaptability.