Certifiable AI Safety Theory (CAST): Convex-Analytic, Measure-Theoretic, and Proof-Carrying Deployment Gates for Tool-Using LLM Systems

CAST (Certifiable AI Safety Theory) provides a rigorous mathematical framework for deployment‑time and runtime safety of modern tool‑using LLM systems. Unlike heuristic safety approaches, CAST formalizes safety as membership of output distributions in closed convex sets, enabling proof‑carrying deployment gates with polynomial‑time verification. The framework addresses a core challenge: LLM systems with tool access (e.g., Python interpreters) generate dependent, adaptive data streams where classical i.i.d. assumptions fail. CAST resolves this using anytime‑valid sequential monitoring based on e‑processes and martingale concentration, yielding time‑uniform guarantees under optional stopping without requiring union bounds across time. Key contributions include: 1. a realizability theorem establishing sound certification with computational feasibility; 2. modular composition rules for composite systems; 3. projection‑to‑safety theorems in KL/Bregman and optimal‑transport geometries; 4. an explicit monitor‑construction layer with architectural guarantees (trusted logging, monitor‑private randomness, deterministic tool replay); 5. global risk budgeting across runtime decision points. CAST targets finance‑grade and scientific‑computing‑grade containment for explicitly specified safety laws, rather than speculative superintelligence scenarios. The framework is presented with full mathematical rigor (theorems, proofs, worked examples) and includes practical guidance on monitor realization, certification protocols, and impossibility results (Rice‑style undecidability, verification trilemma).