The Geometry of Benchmarks: A New Path Toward AGI

Abstract: Benchmarks are the primary tool for assessing progress in AI, yet current practice evaluates models on isolated test suites and provides little guidance for reasoning about generality or autonomous self-improvement. Here we introduce a geometric framework in which all psychometric batteries for AI agents are treated as points in a structured moduli space, and agent performance is described by capability functionals over this space. First, we define an Autonomous AI (AAI) Scale, a Kardashev-style hierarchy of autonomy grounded in measurable performance on batteries spanning families of tasks (for example reasoning, planning, tool use and long-horizon control). Second, we construct a moduli space of batteries, identifying equivalence classes of benchmarks that are indistinguishable at the level of agent orderings and capability inferences. This geometry yields determinacy results: dense families of batteries suffice to certify performance on entire regions of task space. Third, we introduce a general Generator-Verifier-Updater (GVU) operator that subsumes reinforcement learning, self-play, debate and verifier-based fine-tuning as special cases, and we define a self-improvement coefficient $κ$ as the Lie derivative of a capability functional along the induced flow. A variance inequality on the combined noise of generation and verification provides sufficient conditions for $κ> 0$. Our results suggest that progress toward artificial general intelligence (AGI) is best understood as a flow on moduli of benchmarks, driven by GVU dynamics rather than by scores on individual leaderboards.