EULER: Exploring Underused Links with Evidence-Checked Return for Multi-Agent Mathematical Discovery

Mathematical communities work with different objects, invariants, and tools, so transferring a problem across them is expensive and often skipped. We present EULER, a multi-agent system that takes such a transfer--a bridge--as its unit of search. Around a fixed conjecture, EULER runs direct, adjacent-domain, and distant-domain routes in competition; a bridge keeps its budget only if it supplies an operation the source representation cannot execute and its target-side evidence returns to the original statement along a checked implication. Six ordered stress tests reject invalid bridges before expensive search begins. We evaluate EULER on 120 recent conjectures. The conjectures were frozen before search and screened for contamination, and are drawn from public papers by authors who had recently published in the Journal of Combinatorial Theory, Series A, a leading journal in combinatorics. EULER produced 10 proofs and 3 refutations, plus 45 scoped partial results. Two mechanisms held up under ablation: bridge-specific stress tests cut incorrect conclusions from 9 to 3, and bridge material combined with a target-native operation yielded a positive interaction of +4.2 resolved tasks that neither factor produced alone. Domain distance did not reliably predict success; executable operation gain and valid return did.

Autonomous Mathematical Discovery in an Open-World Multi-Agent Environment

We study autonomous mathematical discovery in the Station, an open-world multi-agent environment in which AI agents from different model families pursue a shared research goal without a central coordinator or scripted pipeline. Agents choose their own research directions, conduct experiments, collaborate, and build a shared scientific literature. Across 12 construction problems from the AlphaEvolve catalogue and two additional case studies, the Station obtained results novel relative to the prior literature on five problems: a new infinite family of finite-field Kakeya sets, new exact 604-point kissing configurations in dimension 11, new records for the discretized Kakeya needle and sign uncertainty problems, and a substantially improved lower bound for Erdős's minimum-overlap problem. Agents also discovered novel infinite families for Book Ramsey numbers. Importantly, the agents produced not only numerical constructions but also theorems and analyses explaining how those constructions work, making the results more interpretable and easier for mathematicians to build upon. We release all raw agent dialogues, proofs, and verification code, providing a transparent record of how these discoveries emerged.