Self-evolving Agents

Plane geometry reasoning is a representative testbed for multimodal mathematical intelligence because a solver must jointly interpret a diagram, recover implicit geometric relations, introduce useful auxiliary constructions, and produce logically valid reasoning. Unlike many visual question answering tasks where the image only provides perceptual evidence, geometry problems require the text, diagram, latent constraints, and proof steps to be mutually consistent. A small error in a point relation, angle equality, collinearity constraint, or auxiliary construction can invalidate the entire solution. However, progress in this domain is limited by the scarcity of high-quality multimodal geometry data: constructing valid problems, accurate diagrams, and step-by-step proofs is costly and difficult to scale. Moreover, unlike symbolic mathematical reasoning, visual geometry reasoning lacks easily available verifiable rewards, since correctness depends not only on the final answer but also on whether the diagram, constraints, constructions, and proof are geometrically consistent. These challenges motivate a self-evolving framework that can automatically generate geometry problems, synthesize executable diagrams, train solvers from their own verified experience, and use checkable geometric representations to provide reliable feedback for both data generation and reasoning.