arXiv subjects

Dakai Guo

Publications and source records attributed to Dakai Guo.

5 recordsLinked to original sources

arXiv2026

MechMath Agent Team: LLM Driven Agents for Mathematical Research

AI reasoning has become a central focus in contemporary artificial intelligence, largely driven by the success of large language models. However, mathematical research, which is characterized by non-linear derivation paths, rigorous logical requirements, and protracted exploration cycles, poses severe challenges for existing reasoning systems. To overcome these limitations, we present the MechMath Agent Team (MMAT), which is a large language model driven agent designed to serve as a co-pilot throughout the full cycle of mathematical research. We design a tripartite Harness Architecture that decouples system responsibilities into Control, Execution, and Augmentation planes, thereby reconciling rigorous logical control with the agility demanded by open-ended research. Building upon this framework, we instantiate three specialized agents: a Knowledge Base Manager, a Natural Language Prover, and a Formal Language Prover, all operating in a closed loop to produce formally certified mathematical proofs. We evaluate MMAT on open problems in Number Theory, Algebraic Complexity Theory, Differential Algebra, Operator Algebra, and Inequalities. Across a two-month deployment, 11 problems have been solved, demonstrating its capacity to act as a co-pilot throughout the entire research cycle. The contributions are threefold: a general decoupled Harness Architecture for multi-agent mathematical reasoning, its concrete instantiation in the MMAT system, and empirical validation on a diverse suite of open problems.

cs.AI↗

arXiv2026

Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number

In this paper, it is proved that any nonnegative integer can be written in the following form $$ x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2, \qquad x,y,z \in \mathbb{N}. $$ This settles the conjecture recorded as OEIS A287616. All parts of the proof have been formalized in Lean 4, with the exception of two results: one externally cited theorem and one statement verified by symbolic computation. Both the natural-language proof and the Lean formalization were generated by the MechMath Agent Team developed by the authors.

math.NT↗

arXiv2026

A Greatest Common Divisor Criterion of Certain Binomial Coefficients

The binomial greatest common divisor (gcd) criterion recorded as OEIS A080170 is proven. The criterion also appears as conjecture (17) in Ralf Stephan's list of OEIS conjectures. For $k\geq 2$, put \[ D(k)=\gcd_{2\leq q\leq k+1}\binom{qk}{k}, \qquad n=k+1. \] If $P$ is the largest prime-power component $p^a$ exactly dividing $n$, then the criterion asserts \[ D(k)=1 \quad\Longleftrightarrow\quad \frac{n}{P}>P. \] The proof is formalized in Lean and the Lean artifact is accepted as part of the Formal Conjectures project. Both the natural-language proof and the Lean formalization are generated by the MechMath Agent Team, an AI agent developed by the authors.

math.NT↗

arXiv2026

A Finite Certificate for the Positive $n=9$ Vasc Inequality

We prove the positive-real $n=9$ case of the Vasc cyclic inequality. The proof was obtained with human-guided assistance from the AI agent MechMath Agent Team: the human-readable part reduces the rational inequality to a homogeneous polynomial inequality, fixes a cyclic maximum, and parametrizes each sorted fixed-maximum cone by cumulative gaps; the finite part is a certificate covering all $8!=40320$ sorted cones. MechMath Agent Team generated the certificate verification workflow through Python tool calls, including the case split, verification programs, and terminal classifications. The published certificate has $36815$ coefficient leaves, $2236$ ordinary Polya multiplier leaves, and $1269$ AM-GM midpoint overlay leaves. Human authors audited the mathematical reductions and verification logic, and a separate artifact contains the certificate, an independent verifier, and a from-source rebuild route.

cs.SC↗

arXiv2026

MechMath: Sorrifier-Driven Formal Decomposition Workflow for Automated Theorem Proving

Recent advances in large language models (LLMs) and LLM-based agents have substantially improved the capabilities of automated theorem proving. However, for problems that require complex mathematical reasoning, current systems seldom succeed in their initial attempt, necessitating iterative adjustments to their proof strategies. Existing approaches for handling failed attempts typically either iteratively fix errors within the proof or discard the entire proof and regenerate it from scratch. The former leads to progressively longer contexts, which degrade the model's ability to attend to the remaining unresolved subproblems, while the latter is inefficient, as it may abandon mostly correct reasoning due to localized errors. To address this dilemma, we present MechMath, an agent system centered on a Sorrifier-driven formal decomposition paradigm. By leveraging the sorry placeholder in Lean to precisely isolate unresolved subgoals while preserving the surrounding verified proof structure, MechMath extracts each failed subproblem into a clean, self-contained context and resolves it independently. This avoids both the waste of full regeneration and the excessive context length induced by repeated repairs. Experimental results on challenging mathematical competition benchmarks, including IMO 2025, Putnam 2025, miniF2F, and a subset of ProverBench, demonstrate that our agent achieves significant advantages in proving efficiency.

cs.CL↗