Albilich: Steerable Proof-State Orchestration for LLM-Based Mathematical Research with CAS Integration

Large language models can contribute useful ideas to mathematical research, yet long-horizon proof attempts remain difficult to coordinate, evaluate, and reproduce. We present Albilich, an open-source agentic harness for autoresearch in mathematics that combines long-horizon reasoning, computer algebra systems (CAS), literature retrieval, and persistent SQLite-based context management. We evaluate Albilich on the RealMath benchmark (Zhang et al. 2025) and on open problems in group theory from the Kourovka Notebook (Khukhro and Mazurov 2026). It solved 10/10 problems on RealMath with CAS and 9/10 with no CAS. On the Kourovka problems, Albilich produced a counterexample to Problem 21.142 and a proof of a strengthening of Problem20.2. Anablation on Problem 17.91 demonstrates 32.0% token reduction when CAS is enabled. An ablation on Problem 21.142 demonstrates higher verifier-rejection rate and failure to synthesize proof routes in the absence of the advisor agent. These results support Albilich as a human-steerable, CAS-boosted environment for scalable AI-assisted mathematical research.