ALPS: Measuring Valid Creativity in Large Language Models with Mathematical Construction

We present a new approach for benchmarking Large Language Model (LLM) capabilities on research-level mathematics. Existing benchmarks largely rely on static, hand-curated sets of contest or textbook-style problems as proxies for mathematical research. Instead, we establish an updatable benchmark evaluating models directly on the latest research results in mathematics. This consists of an automatic pipeline that extracts lemmas from arXiv and rewrites them into self-contained statements by making all assumptions and required definitions explicit. It results in a benchmark that can be updated regularly with new problems taken directly from human mathematical research, while previous instances can be used for training without compromising future evaluations. We benchmark current state-of-the-art LLMs, which obtain around 10-15$\%$ accuracy in theorem proving (pass@1) depending on the model, showing that there is currently a large margin of progression for LLMs to reach human-level proving capabilities in a research context.

How should we assess whether large language models can perform mathematical invention? I argue that this question is currently underspecified: mathematical creativity is not one capacity but several mechanistically distinct modes of meaning-making - reflexive introspection on mathematical practice, analogical import from the sciences, problem-driven construction, and the bridging of distant domains - together with a further, cross-cutting distinction between meaning pursued because a pattern was observed and meaning pursued because it is strategically wanted, a distinction I develop through the case of conjecture-formation. These mechanisms are likely non-substitutable, so that competence in one does not transfer to the others. Grounding each in a historical case study and in an architecture-level account of current transformer-based systems, I suggest that today's models concentrate their competence in modes shaped by recombination and search over existing building blocks; if that description holds, the remaining modes are out of reach in principle, not just slower - though whether it holds is itself the open, empirical part. Because proof is getting cheaper as AI improves at generating it - a shift the field's own leading voices are now diagnosing - mathematical value is migrating toward the modes current systems cannot yet perform, and evaluations of AI mathematical ability should be organized around this taxonomy rather than around aggregate benchmarks that conflate it.

Large Language Models (LLMs) are being applied to increasingly difficult problems and use cases. To navigate their vast solution spaces effectively, LLMs need to be creative. Yet the subjective nature of creativity and the limits of human judgment make training LLMs for creativity especially challenging. As a solution, we train LLMs on Codenames, a word-association game that exercises the two central axes of creativity, divergent and convergent thinking, while yielding objectively verifiable outcomes. This verifiability lets us bypass human judgment and train with Reinforcement Learning with Verifiable Rewards (RLVR). We train Qwen3-1.7B, 4B, and 8B models and evaluate them on ten creativity and four reasoning benchmarks. We find that the precision-diversity trade-off is scale-dependent: the 8B model prioritizes creativity over precision, while the 1.7B and 4B models gain reasoning precision at the cost of creativity. Concretely, the 8B model shows modest but consistent creativity gains (8 of 10 benchmarks) with only minor reasoning degradation, whereas the smaller models achieve substantial gains on reasoning tasks. Our study presents a scalable and effective solution to train LLMs for creativity.