A Metric for Interesting Math: Proof Length Over Statement Length
LLMs can now prove theorems that stood open for decades. The next question is less glamorous and more important: out of everything a machine can prove, which theorems are worth having? A paper posted September 23 by Niket Patel, Ahmad Rammal, Amaury Hayat, Remi Munos and Julia Kempe proposes an answer you can compute.
Their definition of intrinsic interestingness is the ratio between how long a theorem's proof is and how long its statement is. Short to state, long to prove, the way the best results tend to look. They show this ratio correlates strongly with an extrinsic measure, how useful a theorem turns out to be downstream. To compute it cheaply they train a 27B model to predict proof difficulty given a set of premises, and it predicts better than frontier general-purpose models do.
Then they optimize for it. A generator trained against the metric produces more interesting theorems, and its overlap with Mathlib, substantial or full, drops from 91.9 percent to 30.6 percent. That means the system stops rediscovering the library it was trained on and starts producing genuinely out-of-distribution mathematics. The loop generates candidates, picks the most interesting, adds them to a self-expanding formal library, and builds on top.
This is the piece the AI-for-math story has been missing. Proof search has a verifier, Lean. Taste did not. Without a taste signal, an autonomous math agent is a machine for generating true and useless statements at scale. A number you can optimize against, even an imperfect one, turns research direction into something an agent loop can hill-climb. Paper: arxiv.org/abs/2609.28603.
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Their definition of intrinsic interestingness is the ratio between how long a theorem's proof is and how long its statement is. Short to state, long to prove, the way the best results tend to look. They show this ratio correlates strongly with an extrinsic measure, how useful a theorem turns out to be downstream. To compute it cheaply they train a 27B model to predict proof difficulty given a set of premises, and it predicts better than frontier general-purpose models do.
Then they optimize for it. A generator trained against the metric produces more interesting theorems, and its overlap with Mathlib, substantial or full, drops from 91.9 percent to 30.6 percent. That means the system stops rediscovering the library it was trained on and starts producing genuinely out-of-distribution mathematics. The loop generates candidates, picks the most interesting, adds them to a self-expanding formal library, and builds on top.
This is the piece the AI-for-math story has been missing. Proof search has a verifier, Lean. Taste did not. Without a taste signal, an autonomous math agent is a machine for generating true and useless statements at scale. A number you can optimize against, even an imperfect one, turns research direction into something an agent loop can hill-climb. Paper: arxiv.org/abs/2609.28603.
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