Current paradigms in Artificial Intelligence face a seemingly intractable trade-off between Capability and Safety. Systems capable of recursive self-improvement (the "Singularity") risk unbounded instability, while systems with provable safety guarantees ("Bounded Optimality") are mathematically prevented from generating novel solutions. This paper introduces the General Interaction Dynamics Engine (GIDE), a theoretical framework that resolves this paradox. We present a mechanism for Finite Recursive Growth—an "Intelligence Ratchet"—that allows an AI system to exhibit transient bursts of superintelligent creativity while remaining rigorously bounded by physical and information-theoretic safety constraints. We formally define the Predictability Horizon ($T_{\text{pred}}$) for chaotic cognition and the Semantic Grounding Invariant that couples learning rates to physical verification.