Andy: A Mathematical Agent for Rigorous Proof and Autonomous Research

This paper investigates the autonomous synchronization problem for discrete-time heterogeneous multiagent systems. The synchronization problem is transformed into the asymptotic decoupling problem of stable modes in a class of discrete-time linear time-varying systems, for which we provide a sufficient condition. Leveraging this condition, synchronization conditions are established. The synchronization conditions are based on the average of the agents' initial dynamic matrices, without requiring the differences among these matrices to be small. This approach reduces the conservativeness of existing conditions and achieves a unification of both homogeneous and heterogeneous systems. Numerical simulation results are provided to support the theoretical findings.

Mathematical concepts emerge through an interplay of processes, including experimentation, efforts at proof, and counterexamples. In this paper, we present a new multi-agent model for computational mathematical discovery based on this observation. Our system, conceived with research in mind, poses its own conjectures and then attempts to prove them, making decisions informed by this feedback and an evolving data distribution. Inspired by the history of Euler's conjecture for polyhedra and an open challenge in the literature, we benchmark with the task of autonomously recovering the concept of homology from polyhedral data and knowledge of linear algebra. Our system completes this learning problem. Most importantly, the experiments are ablations, statistically testing the value of the complete dynamic and controlling for experimental setup. They support our main claim: that the optimisation of the right combination of local processes can lead to surprisingly well-aligned notions of mathematical interestingness.

Long-horizon mathematical reasoning fails less often because a model cannot produce a valid next step than because an agent fails to maintain and expose the right semantic state across many iterations. Left unmanaged, this produces research-level proofs that are locally convincing yet globally incomplete: a key lemma unproved, an assumption unchecked, a citation unsupported, or a computational claim unverified. We present Research Math Agents (RMA), an agentic framework for long-horizon proof development built around a persistent, typed research store and an orchestrator that compiles operation-specific context from that store. The Research Context Orchestrator is the central state-management layer between the persistent research store and each locally scoped proof operation: it retrieves task-relevant artifacts, compiles them into a bounded context, invokes the appropriate operation, and writes the resulting proof edits, issue updates, literature notes, plans, or evaluations back to the store. This process is designed to keep proof revisions, unresolved issues, prior attempts, literature, and evaluations available across rounds while exposing only task-relevant state to each local operation. We evaluate RMA across complementary research-level settings using independent expert evaluation, blind mathematician review, LLM-based benchmark evaluation, and Lean 4 kernel verification. RMA achieves a 42.5% solve rate on the independently evaluated SOOHAK Challenge Hard set, obtains 8 of 10 correct solutions on First Proof B1 and 8 of 10 passing solutions on B2 under human-expert evaluation, and verifies 213 of 300 sampled Research Solved targets in Formal Conjectures with the Lean 4 kernel.