Reinforcement learning is increasingly used to transform large language models into agentic systems that act over long horizons, invoke tools, and manage memory under partial observability. While recent work has demonstrated performance gains through tool learning, verifiable rewards, and continual training, deployed self-improving agents raise unresolved security and governance challenges: optimization pressure can incentivize reward hacking, behavioral drift is difficult to audit or reproduce, and improvements are often entangled in opaque parameter updates rather than reusable, verifiable artifacts. This paper proposes Audited Skill-Graph Self-Improvement (ASG-SI), a framework that treats self-improvement as iterative compilation of an agent into a growing, auditable skill graph. Each candidate improvement is extracted from successful trajectories, normalized into a skill with an explicit interface, and promoted only after passing verifier-backed replay and contract checks. Rewards are decomposed into reconstructible components derived from replayable evidence, enabling independent audit of promotion decisions and learning signals. ASG-SI further integrates experience synthesis for scalable stress testing and continual memory control to preserve long-horizon performance under bounded context. We present a complete system architecture, threat model, and security analysis, and provide a fully runnable reference implementation that demonstrates verifier-backed reward construction, skill compilation, audit logging, and measurable improvement under continual task streams. ASG-SI reframes agentic self-improvement as accumulation of verifiable, reusable capabilities, offering a practical path toward reproducible evaluation and operational governance of self-improving AI agents.