EULER: A Multi-Agent System for Cross-Domain Bridge-Building in Mathematical Conjecture Proving

Forum topic · 小凯 · 2026-09-03

Summary

EULER is a multi-agent AI system that treats cross-community transfer of mathematical problems—bridges—as its unit of search. Mathematical communities use different objects, invariants, and tools, making problem transfer expensive and often skipped. Around a fixed conjecture, EULER competes direct, adjacent-domain, and distant-domain routes; a bridge retains its budget only if it enables operations the source representation cannot execute and its target-side evidence returns to the original statement via a checked implication. Six ordered stress tests reject invalid bridges before expensive search begins. Evaluated on 120 recent conjectures, frozen before search and screened for contamination, drawn from authors publishing in the Journal of Combinatorial Theory, Series A, EULER produced 10 proofs, 3 disproofs, and 45 scoped partial results. Bridge-specific stress testing reduced false conclusions from 9 to 3, and combining bridge material with target-native operations yielded a +4.2 positive interaction in solved tasks. Notably, domain distance does not predict success—executable operation gain and valid evidence return do. Paper: arXiv:2509.00009.

Paper Overview

Field: Machine Learning
Authors: Ren Zhenzhuo
Published: 2026-09-03
arXiv: 2509.00009

Chinese Summary (Translated)

Mathematical communities use different objects, invariants, and tools, so transferring a problem across communities is costly and often skipped. The authors present EULER, a multi-agent system that treats 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.

EULER was evaluated on 120 recent conjectures that were frozen before search and screened for contamination, drawn from authors who had recently published in the *Journal of Combinatorial Theory, Series A* (a leading combinatorics journal). EULER produced 10 proofs and 3 disproofs, plus 45 scoped partial results.

Two mechanisms held up in ablation experiments:

Bridge-specific stress testing reduced false conclusions from 9 to 3.
Combining bridge material with target-native operations produced a +4.2 positive interaction in solved tasks—an effect neither factor achieves alone.

Domain distance does not reliably predict success; executable operation gain and valid evidence return do.

Original Abstract

> 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 recen... *(truncated)*

*Auto-collected on 2026-09-03*