A Mathematical Solution to the AI Alignment Problem: Topological Constraints on Action Distributions with Progressive Verification

This paper presents a first-principles mathematical framework for AI alignment, decoupling safety from training data quality through topological constraints on action distributions and progressive verification. We model deployed AI behavior as probability measures over infinite-dimensional trajectory spaces, enforcing alignment as membership in a closed feasible set defined by risk functionals. Key contributions include existence theorems for aligned projections under Wasserstein regularity, operationalization via LTL and sequential drift detection, conformal/PAC certification for finite logs, and extensions to multi-agent equilibria. Scoped to information-work systems, the approach emphasizes external, auditable guarantees compatible with governance regimes like NIST AI RMF and EU AI Act. Includes LaTeX source, appendices on LTL operationalization and structured interaction spaces for specification relief.