A Mathematical Framework for AI Singularity: Conditions, Bounds, and Control of Recursive Improvement

摘要

AI systems improve by using more compute, data, and better algorithms. A common fear is that this improvement might feed on itself and run away—what people call a “singularity” [18]. This paper asks a precise version of that question: under what measurable conditions could capability growth run away in finite time, and under what conditions can we rule that out? We aim for answers that are testable with real power, bandwidth, and benchmark data, and for controls that can be implemented in practice (e.g., power caps, throttling, evaluation gates). The goal is to replace speculation with clear conditions that either certify safety or flag genuine risk—without relying on simulations or unverifiable assumptions.