EULER: Searching Representation–Operation Pairs for Mathematical Discovery

A change of representation can make a mathematical problem accessible to an operation that was previously unavailable. The challenge is to choose that change and return valid conclusions to the original problem. We introduce EULER, a multi-agent system that searches representation–operation pairs alongside direct proof attempts. Each candidate records a map, an enabled operation, and the conditions for returning a theorem or counterexample to the source problem. On 120 source tasks, the system averages 12.8 resolutions over five reasoning seeds. A matched $2\times2$ experiment finds an interaction of 4.2 resolutions between supplied bridge material and target-native tool access (95% interval $1.0,7.4$). EULER autonomously completed ten mathematical results end to end on conjectures posed by authors who had published in JCTA within the preceding 24 months. Its zero-sum investigation connects counterexamples to Zhao's conjectures with the exact formula $E(\mathrm{Dih}(A))=2|A|+D(A)$ for nontrivial finite abelian odd $p$-groups $A$. The experiments test bridge use and selection; the proofs demonstrate autonomous mathematical execution.