Large language models have made striking progress on advanced mathematics, solving problems that had remained open for decades. According to a new paper posted to Hugging Face, that success raises a fresh question: even if models can conjecture and prove more theorems, is the resulting mathematics actually interesting or useful?
The authors propose defining the intrinsic interestingness of a theorem as the ratio between the length of its proof and the length of its statement. They report that this measure correlates strongly with an extrinsic measure of a theorem's downstream utility.
To compute these metrics, the work identifies the difficulty of a proof conditioned on a set of premises as a useful primitive. The team trained a 27B model that predicts proof difficulty more accurately than frontier general-purpose models, according to the paper.
Optimizing for the metric produced a model that generates more interesting theorems while cutting substantial or full overlap with Mathlib from 91.9% to 30.6% — which the authors present as evidence of more out-of-distribution mathematics.
The paper describes a system that can generate candidate theorems, select the most interesting among them, and iteratively build on a self-expanding mathematical library. The authors frame these metrics as a practical, quantifiable signal for ranking conjectures and guiding proof search in formal libraries, and as a path toward self-expanding, machine-verified mathematical libraries that can choose worthwhile statements without human-supplied targets.