From Optimality to Viability: Survival-Satisficing, Proof-Carrying Gödelian Language Models with Evolutionary Self-Adaptation

This paper introduces a survival-satisficing Gödelian learning machine that evolves self-modifying language-model progeny by prioritizing long-term viability, bounded operational well-being, and catastrophic-risk control over unconstrained utility maximization. The framework combines proof-carrying invariants, Lean/SMT verification, robust and risk-sensitive learning, evolutionary bet-hedging, and anytime-valid statistical certification, while adapting persistent self-edit mechanisms from recent self-adapting language models. By formalizing what it means for an LLM to be Gödelian and providing a reproducible architecture, algorithms, guarantees, and implementation sketches, the work offers a rigorous foundation for safer recursive self-improvement under uncertainty.