Hanzhang Liu

Linbin Tang, Jingyan You, Zilin Kang et al.

Recent formal reasoning systems have reached IMO-level performance, yet they leave a fragmented landscape: algebra and number theory are handled in Lean, while geometry still relies on domain-specific languages with limited formal guarantees. This split increases the trusted computing base and hinders unified model development. Existing geometry-in-Lean efforts (LeanEuclid, LeanGeo) introduce custom axiom systems incompatible with standard Mathlib, and their small scale ($<$ 1,100 problems) limits large-scale training. Native Mathlib autoformalization of geometry, however, poses distinct challenges: implicit diagrammatic assumptions (e.g., topological configuration and non-degeneracy) must be made explicit rather than deferred to external solvers, and models must adapt to Mathlib's small, rapidly evolving geometry infrastructure. We present Euclean, a four-stage framework - constraint explication, configuration anchoring, formalization mapping, and iterative repair - for automatically formalizing geometry in native Mathlib. We construct OMNI-Geometry (768 competition problems) and Numina-Geometry (177,597 problems), the largest geometry formalization dataset in Lean. Human evaluation shows 48.89% TOP1 and 73.33% TOP5 accuracy. Training Goedel v2 on our formalizations improves proof success from 13.6% to 15.1%, validating dataset quality for unified neural theorem proving. Code and datasets: https://github.com/tlb-22/Euclean.

Jiaao Wu, Xian Zhang, Hanzhang Liu et al.

Frontier AI models and multi-agent systems have led to significant improvements in mathematical reasoning. However, for problems requiring extended, long-horizon reasoning, existing systems continue to suffer from fundamental reliability issues: hallucination accumulation, memory fragmentation, and imbalanced reasoning-tool trade-offs. In this paper, we introduce STAR-PólyaMath, a multi-agent framework that systematically addresses these challenges through meta-level supervision and structured Reasoner-Verifier interaction. STAR-PólyaMath is structured as an orchestrated state machine with nested challenge-step-replan loops, governed by a reasoning-free Python orchestrator that separates control from inference and bounds error propagation through trace-back and re-planning. Our key innovation is a persistent Meta-Strategist that maintains cross-attempt memory and exercises meta-level control by issuing high-level strategic guidance or mandatory directives, so the system can escape unproductive loops rather than stagnate or over-rely on tools. STAR-PólyaMath achieves state-of-the-art results on all eight top-tier competition benchmarks: AIME 2025-2026, MathArena Apex Shortlist, MathArena Apex 2025, Putnam 2025, IMO 2025, HMMT February 2026, and USAMO 2026. It obtains perfect scores on AIMEs, Putnam, and HMMT, and shows its largest margin on Apex 2025, scoring 93.75% compared with 80.21% by the strongest baseline GPT-5.5. Ablation studies show that the gains arise from the framework's orchestration rather than from model-level diversity since removing key components or substituting in mixed backbones consistently weakens performance. Code is available at https://github.com/Julius-Woo/STAR-PolyaMath.