LungClaw introduces a deterministic metabolic governance engine for autonomous agentic systems. As AI agents transition from assistance to execution, traditional orchestration models fail to provide bounded cost, policy enforcement, and runtime stability. This paper formalizes LungClaw’s governance model through a constraint-based metabolic control framework. The engine enforces deterministic transition validation, bounded execution cost propagation, and non-adaptive policy containment. We define formal invariants preventing retry amplification, unbounded execution loops, and cross-agent escalation. LungClaw guarantees atomic energy commits, monotonic constraint evaluation, and fail-safe interruption semantics. This work positions LungClaw as a foundational governance layer for large-scale autonomous infrastructures and agentic economic systems.