Reactivating Test-Time Scaling for Plane Geometry Problem Solving

Abstract:Plane geometry problem (PGP) solving has become a critical benchmark for multimodal reasoning because it requires accurate visual perception and precise multi-step symbolic deduction. Although test-time scaling (TTS) has demonstrated remarkable success in general mathematical reasoning, it fails to scale effectively under the symbolic-program paradigm for plane geometry. We identify two key obstacles: limited reasoning diversity induced by rigid symbolic programs and insufficient explicit visual grounding before symbolic deduction. To address these issues, we propose Multi-Trace Synthesis (MTS), which converts each symbolic program into heterogeneous reasoning traces, including executable Python scripts and CoT-augmented variants. We further propose Perception-Augmented (PA) training, which parses diagrams into structured semantic clauses before deduction, and Consensus-Guided Multi-Trace Ensemble (CG-MTE) for efficient self-adaptive inference. Experiments on three geometry benchmarks show that our method consistently improves PGP-solving across model scales and achieves strong performance against both general-purpose MLLMs and specialized geometry solvers. Under test-time scaling, CG-MTE achieves comparable accuracy to high-budget self-consistency while reducing sampling cost by up to 8x. Code and data are publicly available at this https URL.

Computation and Language (cs.CL)

arXiv:2608.30156 [cs.CL]

(or arXiv:2608.30156v1 [cs.CL] for this version)

https://doi.org/10.48550/arXiv.2608.30156 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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