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.

Mechanic: 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 requiring complex mathematical reasoning, current systems rarely succeed on the first try and must repeatedly modify their proof strategies. Existing approaches for handling failed attempts typically either discard the entire proof and regenerate it from scratch or iteratively fix errors within the proof. The former is inefficient, as it may abandon mostly correct reasoning due to localized errors, while the latter, although preserving prior progress, leads to progressively longer contexts which progressively degrades the model's ability to attend to the remaining unresolved subproblems. To address this dilemma, we propose Mechanic, a novel agent system that employs a sorry-driven formal decomposition strategy. By leveraging the sorry placeholder in Lean to precisely isolate unresolved subgoals while preserving the surrounding verified proof structure, Mechanic 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 and Putnam 2025, demonstrate that our agent achieves significant advantages in proving efficiency.

PowerMLP: An Efficient Version of KAN

The Kolmogorov-Arnold Network (KAN) is a new network architecture known for its high accuracy in several tasks such as function fitting and PDE solving. The superior expressive capability of KAN arises from the Kolmogorov-Arnold representation theorem and learnable spline functions. However, the computation of spline functions involves multiple iterations, which renders KAN significantly slower than MLP, thereby increasing the cost associated with model training and deployment. The authors of KAN have also noted that ``the biggest bottleneck of KANs lies in its slow training. KANs are usually 10x slower than MLPs, given the same number of parameters.'' To address this issue, we propose a novel MLP-type neural network PowerMLP that employs simpler non-iterative spline function representation, offering approximately the same training time as MLP while theoretically demonstrating stronger expressive power than KAN. Furthermore, we compare the FLOPs of KAN and PowerMLP, quantifying the faster computation speed of PowerMLP. Our comprehensive experiments demonstrate that PowerMLP generally achieves higher accuracy and a training speed about 40 times faster than KAN in various tasks.

Unlink to Unlearn: Simplifying Edge Unlearning in GNNs

As concerns over data privacy intensify, unlearning in Graph Neural Networks (GNNs) has emerged as a prominent research frontier in academia. This concept is pivotal in enforcing the right to be forgotten, which entails the selective removal of specific data from trained GNNs upon user request. Our research focuses on edge unlearning, a process of particular relevance to real-world applications, owing to its widespread applicability. Current state-of-the-art approaches like GNNDelete can eliminate the influence of specific edges, yet our research has revealed a critical limitation in these approaches, termed over-forgetting. It occurs when the unlearning process inadvertently removes excessive information beyond specific data, leading to a significant decline in prediction accuracy for the remaining edges. To address this issue, we have identified the loss functions of GNNDelete as the primary source of the over-forgetting phenomenon. Furthermore, our analysis also suggests that loss functions may not be essential for effective edge unlearning. Building on these insights, we have simplified GNNDelete to develop Unlink-to-Unlearn (UtU), a novel method that facilitates unlearning exclusively through unlinking the forget edges from graph structure. Our extensive experiments demonstrate that UtU delivers privacy protection on par with that of a retrained model while preserving high accuracy in downstream tasks. Specifically, UtU upholds over 97.3% of the retrained model's privacy protection capabilities and 99.8% of its link prediction accuracy. Meanwhile, UtU requires only constant computational demands, underscoring its advantage as a highly lightweight and practical edge unlearning solution.