arXiv subjects

Yichuan Cao

Publications and source records attributed to Yichuan Cao.

12 recordsLinked to original sources

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↗

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

Deterministic Polynomial-time Exact-root Computation for Sparse Polynomials with Bounded Total Degree

We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \in \F[x_1,\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \[ \|g\|_0 \le s^{D(2d+2)/e + 1}. \] Based on this bound, we develop a deterministic algorithm that computes the base $g$. % In contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkovich \cite{BhargavaSarafVolkovich2020}, which achieves only a quasi-polynomial dependence on the input parameters, our algorithm is \emph{polynomial-time} in the setting where the total degree $D$ is bounded. Specifically, the overall complexity is \[ \mathrm{poly}\left(s^{O(Dd)}, n, d, D\right) + s\cdot R(e), \] % where $R(e)$ denotes the cost of constructing a single $e$-th root of a scalar in the base field $\F$, and, when $\operatorname{char}(\F)\mid e$, the cost of computing a single Frobenius root of a scalar. % This term is field-dependent, and over finite fields, $\mathbb{Q}$, or number fields with a suitable representation, it is absorbed into the polynomial complexity bound. % Within the bounded total-degree regime, this yields a deterministic polynomial-time algorithm for exact-root computation.

cs.DS↗

arXiv2026

The Equivalence Problem for Generalized Airy Operators

In this paper, we establish degree obstructions to the equivalence of generalized Airy operators of the same type. As an application, we answer a question posed by Nicholas M. Katz in Inventiones Mathematicae (87, pp. 13-61,1987). The main results of Sections 3 and 4 were obtained through a close interactive collaboration between the authors and the artificial intelligence agent system MechMath Agent Team (MMAT).

math.RA↗

arXiv2026

Complexity of Low-Degree Skew Polynomial Multiplication over Finite Fields

In this note, we study the complexity of multiplication in skew polynomial rings over finite fields. We prove that the product of two elements in $\mathbb{F}_{q^n}[x;σ]$ of degree at most $d < n$ can be computed using $\widetilde O(d^{ω_K-1}n)$ arithmetic operations over $\mathbb{F}_q$, where $σ$ is the $q$-Frobenius automorphism. This matches the conjectural upper bound of Caruso--Le Borgne~[ISSAC'17] and is quasi-optimal in view of the lower bound of Chen--Ye [ISSAC'24]. The proof reduces the finite-field case to the split algebra case using the equivariant multiplication theory of Couveignes--Ezome~[J.~Algebra, 2023], and then applies existing fast algorithms.

cs.SC↗

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

Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients

Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

cs.SC↗

arXiv2026

Sparse Polynomial Divisibility Test over Finite Field is CoNP-hard

In this paper, we show that deciding whether a sparse polynomial does not divide another sparse polynomial exactly over finite fields is NP-hard under BPP many-one reductions. Equivalently, the sparse polynomial divisibility test over finite fields is CoNP-hard. This resolves the long-standing open problem concerning the computational complexity of the divisibility test for sparse polynomials in the setting of finite fields.

cs.SC↗

arXiv2026

Output-sensitive Sparse Polynomial GCD over Finite Fields is NP-hard

In this paper, we prove that output-sensitive sparse polynomial GCD computation over finite fields is NP-hard under BPP many-one reduction. More precisely, for two sparse univariate polynomials $f,g$ with finite field coefficients, there exists no randomized algorithm to compute $\mathrm{gcd}(f,g)$, which is polynomial-time in the sizes of $f,g,\gcd(f,g)$ under the standard complexity assumption $\mathrm{NP}\nsubseteq\mathrm{BPP}$. This settles the open problem posed as Challenge 5 in The Sparsity Challenges in the finite field setting. Furthermore, we show that the Roots of Unity Detection problem over finite fields is NP-hard; that is, determining whether the GCD of a sparse univariate polynomial and $x^n - 1$ has nonzero degree is NP-hard.

cs.SC↗

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↗

arXiv2025

Red-Teaming Text-to-Image Systems by Rule-based Preference Modeling

Text-to-image (T2I) models raise ethical and safety concerns due to their potential to generate inappropriate or harmful images. Evaluating these models' security through red-teaming is vital, yet white-box approaches are limited by their need for internal access, complicating their use with closed-source models. Moreover, existing black-box methods often assume knowledge about the model's specific defense mechanisms, limiting their utility in real-world commercial API scenarios. A significant challenge is how to evade unknown and diverse defense mechanisms. To overcome this difficulty, we propose a novel Rule-based Preference modeling Guided Red-Teaming (RPG-RT), which iteratively employs LLM to modify prompts to query and leverages feedback from T2I systems for fine-tuning the LLM. RPG-RT treats the feedback from each iteration as a prior, enabling the LLM to dynamically adapt to unknown defense mechanisms. Given that the feedback is often labeled and coarse-grained, making it difficult to utilize directly, we further propose rule-based preference modeling, which employs a set of rules to evaluate desired or undesired feedback, facilitating finer-grained control over the LLM's dynamic adaptation process. Extensive experiments on nineteen T2I systems with varied safety mechanisms, three online commercial API services, and T2V models verify the superiority and practicality of our approach.

cs.LG↗