arXiv subjects

Yunzhou Xie

Publications and source records attributed to Yunzhou Xie.

5 recordsLinked to original sources

arXiv2026

Numina-Lean-Agent: An Open and General Agentic Reasoning System for Formal Mathematics

Agentic systems have recently become the dominant paradigm for formal theorem proving, achieving strong performance by coordinating multiple models and tools. However, existing approaches often rely on task-specific pipelines and trained formal provers, limiting their flexibility and reproducibility. In this paper, we propose the paradigm that directly uses a general coding agent as a formal math reasoner. This paradigm is motivated by (1) A general coding agent provides a natural interface for diverse reasoning tasks beyond proving, (2) Performance can be improved by simply replacing the underlying base model, without training, and (3) MCP enables flexible extension and autonomous calling of specialized tools, avoiding complex design. Based on this paradigm, we introduce Numina-Lean-Agent, which combines Claude Code with Numina-Lean-MCP to enable autonomous interaction with Lean, retrieval of relevant theorems, informal proving and auxiliary reasoning tools. Using Claude Opus 4.5 as the base model, Numina-Lean-Agent solves all problems in Putnam 2025 (12 / 12), matching the best closed-source system. Beyond benchmark evaluation, we further demonstrate its generality by interacting with mathematicians to successfully formalize the Brascamp-Lieb theorem. We release Numina-Lean-Agent and all solutions at https://github.com/project-numina/numina-lean-agent.

cs.AI↗

arXiv2025

Cocycle weighted infinitesimal bialgebras and pre-Lie algebras on rooted trees

The concept of weighted infinitesimal bialgebras provides an algebraic framework for understanding the non-homogeneous associative Yang-Baxter equation. In this paper, we endow the space of decorated planar rooted forests with a two-parameters family of coproducts, making it into a weighted infinitesimal bialgebra. A combinatorial characterization of the coproducts is given via the notion of forest biideals. Furthermore, by constructing a bilinear symmetric form and introducing a new grafting operation on rooted forests, we describe the associated dual products. We also introduce the notion of the pair-weight 1-cocycle condition and investigate the universal properties of decorated planar rooted forests satisfying this condition. This leads to the definition of a weighted $Ω$-cocycle infinitesimal unitary bialgebra. As applications, we identify the initial object in the category of free cocycle infinitesimal unitary bialgebras on undecorated planar rooted forests, corresponding to the well-known noncommutative Connes-Kreimer Hopf algebra. In addition, we establish isomorphisms between different coproduct structures and construct a pre-Lie algebra structure on decorated planar rooted forests.

math.CO↗

arXiv2025

Moerdijk Hopf algebras of decorated rooted forests: an operated algebra approach

In this paper, we first endow the space of decorated planar rooted forests with a coproduct that equips it with the structure of a bialgebra and further a Moerdijk Hopf algebra. We also present a combinatorial description of this coproduct, and further give an explicit formulation of its dual coproducts through the newly defined notion of forest-representable matrices. By viewing the Moerdijk Hopf algebra within the framework of operated algebras, we introduce the notion of a multiple cocycle Hopf algebra, incorporating a symmetric Hochschild 1-cocycle condition. We then show that the antipode of this Hopf algebra is a Rota-Baxter operator on Moerdijk Hopf algebras. Furthermore, we investigate the universal properties of cocycle Hopf algebras. As an application, we construct the initial object in the category of free cocycle Hopf algebras on undecorated planar rooted forests, which coincides with the well-known Moerdijk Hopf algebra.

math.RA↗

arXiv2025

CombiBench: Benchmarking LLM Capability for Combinatorial Mathematics

Neurosymbolic approaches integrating large language models with formal reasoning have recently achieved human-level performance on mathematics competition problems in algebra, geometry and number theory. In comparison, combinatorics remains a challenging domain, characterized by a lack of appropriate benchmarks and theorem libraries. To address this gap, we introduce CombiBench, a comprehensive benchmark comprising 100 combinatorial problems, each formalized in Lean~4 and paired with its corresponding informal statement. The problem set covers a wide spectrum of difficulty levels, ranging from middle school to IMO and university level, and span over ten combinatorial topics. CombiBench is suitable for testing IMO solving capabilities since it includes all IMO combinatorial problems since 2000 (except IMO 2004 P3 as its statement contain an images). Furthermore, we provide a comprehensive and standardized evaluation framework, dubbed Fine-Eval (for $\textbf{F}$ill-in-the-blank $\textbf{in}$ L$\textbf{e}$an Evaluation), for formal mathematics. It accommodates not only proof-based problems but also, for the first time, the evaluation of fill-in-the-blank questions. Using Fine-Eval as the evaluation method and Kimina Lean Server as the backend, we benchmark several LLMs on CombiBench and observe that their capabilities for formally solving combinatorial problems remain limited. Among all models tested (none of which has been trained for this particular task), Kimina-Prover attains the best results, solving 7 problems (out of 100) under both ``with solution'' and ``without solution'' scenarios. We open source the benchmark dataset alongside with the code of the proposed evaluation method at https://github.com/MoonshotAI/CombiBench/.

cs.AI↗

arXiv2025

Species of Rota-Baxter algebras by rooted trees, twisted bialgebras and Fock functors

As a fundamental and ubiquitous combinatorial notion, species has attracted sustained interest, generalizing from set-theoretical combinatorial to algebraic combinatorial and beyond. The Rota-Baxter algebra is one of the algebraic structures with broad applications from Renormalization of quantum field theory to integrable systems and multiple zeta values. Its interpretation in terms of monoidal categories has also recently appeared. This paper studies species of Rota-Baxter algebras, making use of the combinatorial construction of free Rota-Baxter algebras in terms of angularly decorated trees and forests. The notion of simple angularly decorated forests is introduced for this purpose and the resulting Rota-Baxter species is shown to be free. Furthermore, a twisted bialgebra structure, as the bialgebra for species, is established on this free Rota-Baxter species. Finally, through the Fock functor, another proof of the bialgebra structure on free Rota-Baxter algebras is obtained.

math.CO↗