SynGeo: Synergizing Seeing and Proving through Revisable Geometric States

Geometry problem solving is a canonical testbed for machine intelligence, requiring systems to interpret diagrams, ground symbolic constraints, and perform rigorous deduction. Yet current approaches expose a persistent gap between seeing and proving: multimodal large language models can flexibly inspect diagrams but may hallucinate unsupported relations, while symbolic solvers provide checkable derivations but are brittle to incomplete or misgrounded formalization. We introduce SynGeo, a state-centric framework for synergizing seeing and proving in geometry problem solving by making the geometric representation revisable during inference. SynGeo first constructs a predicate state from diagram--text evidence and tests it with symbolic reasoning. When proof search fails or stagnates, symbolic diagnostics guide image-grounded revisiting, repairing solver-incompatible predicates before another deductive attempt. When the symbolic route remains unresolved, a complementary MLLM branch provides an image-grounded reasoning path from the problem evidence. On Geometry3K and PGPS9K, SynGeo achieves state-of-the-art performance across both Choice and Completion settings. With GPT-4o as the backbone, it reaches (90.2\%) accuracy in both settings on Geometry3K, and 90.1% Choice accuracy and 88.6% Completion accuracy on PGPS9K. Ablations show that both feedback-guided revisiting and complementary reasoning are necessary, supporting a broader view of geometry problem solving as an adaptive loop between what a system sees, what it formalizes, and what it proves.