Learning to Discover Interesting Mathematics

TL;DR

Every AI-for-mathematics result this wiki holds is scored against problems humans chose. This paper asks what happens when nobody supplies the target, and answers with a metric: the intrinsic interestingness of a theorem is the ratio of its proof length to its statement length. That ratio is shown to correlate strongly with an extrinsic measure of downstream utility. A 27B model trained to predict proof difficulty conditioned on a premise set does so more accurately than frontier general-purpose models, and optimising for the metric cuts substantial-or-full overlap with Mathlib from 91.9% to 30.6% — the system stops rediscovering the library it was given (source).

Authors & Org

Niket Patel, Ahmad Rammal, Amaury Hayat, Rémi Munos, Julia Kempe (2 search passes agree on the list). Affiliations, from the first pass and consistent with the second: FAIR @ Meta (Patel, Rammal, Munos, Kempe), CERMICS, ENPC, Institut Polytechnique de Paris (Rammal, Hayat), New York University (Kempe). arxiv.org answers EGRESS_BLOCKED from this run's sandbox and the HuggingFace snapshot carries no author block, so the affiliation split is second-hand and recorded as such. Submitted 2026-09-23.

This is Meta AI's FAIR, on formal mathematics, with Rémi Munos — an RL author — on the byline.

Method

The paper's structure is three claims stacked, and the order matters because each one is only useful if the previous one holds:

1. Define interestingness without a human. Intrinsic interestingness:= proof length / statement length. A theorem that says little and costs a lot to prove scores high; a restatement of a definition scores low.
2. Show the proxy is not circular. The ratio is shown to correlate strongly with an extrinsic measure of a theorem's downstream utility — how much later work the theorem is actually used by. This is the load-bearing step: without it the metric measures verbosity.
3. Make it computable. Both quantities reduce to the difficulty of a proof conditioned on a set of premises, identified as the useful primitive. A 27B model is trained to predict that difficulty, and beats frontier general-purpose models at it.

With the primitive in hand, the system runs a loop: generate candidate theorems → select the most interesting → add them to the library → build on the expanded library. The library is machine-verified, so the selection pressure operates on statements that are already known to be true.

Results

Measurement	Value
Difficulty predictor	27B, more accurate than frontier general-purpose models
Mathlib overlap, before optimising for the metric	91.9% substantial or full
Mathlib overlap, after	30.6%
Intrinsic↔extrinsic relationship	strong correlation (coefficient not in the snapshot)

• The 91.9% → 30.6% figure is the result. It is a measure of novelty relative to an existing library, not of correctness or of value: every theorem on both sides of it is machine-verified. What it says is that a system generating theorems without the metric spends most of its output re-deriving Mathlib, and that optimising for proof-to-statement ratio pushes it out of distribution — which the paper names as the point.
• The 27B model beating frontier models is a specialisation result, on one narrow prediction task (proof difficulty given premises), and the snapshot does not name the frontier models it was compared against.
• The system is shown to run the full loop — generate, select, extend, build on its own additions — as a demonstration. No figure is given for how far the loop was run or what the library looked like at the end.

Significance

AI for Mathematics carries four dated State-of-the-Art sections, and each of them measures a model against a problem somebody set: competition problems, open conjectures, formalisation throughput, Astra's ten advances. The selection of what to work on has been human in every one of them. This paper attacks that layer directly, and it does so with a metric rather than a taste judgement — which is why it belongs on that page rather than in a list of capability claims.

Why the ratio is a good idea and where it breaks. A statement-to-proof ratio is cheap, objective and gameable in a known direction: it rewards short statements with long proofs, which is a fair description of a deep theorem and also a fair description of an obfuscated one. The paper's defence is the correlation with downstream utility, and that correlation is doing all the work — it is the only thing standing between "interesting" and "expensive to verify". The snapshot states the correlation is strong and gives no coefficient, so the defence is asserted here and not quantified.

For R&D Automation Index the connection is narrower than it looks. That page tracks automation of research the field already wanted done. This is automation of wanting — choosing the target — and the honest reading is that it is demonstrated inside a formal library, where truth is decidable and a premise set is enumerable. Nothing here transfers to a domain where a conjecture cannot be machine-checked, and the paper does not claim it does.

Read against Coding Agents for Generalized Task and Motion Planning Problems, captured the same day: both take a job that had been human — engineering a planner, choosing a theorem — and hand it to a model with a verifier in the loop. Neither works without the verifier.

Open Questions

• What is the correlation coefficient? "Strong" is the whole argument for the metric and the number is not in the snapshot.
• Which frontier models did the 27B beat, on what difficulty benchmark, and by how much? None of the three is stated.
• Is 30.6% overlap good? It is lower, which is what was optimised for. Whether the 69.4% that is not in Mathlib is mathematics anyone wants is exactly the question the extrinsic measure is supposed to answer, and no extrinsic score is reported for the generated theorems — only for the correlation study.
• Does the ratio survive adversarial statements? Nothing read tests whether a model can inflate proof length without inflating depth.
• What is the base model for the 27B difficulty predictor, and what was it trained on? Not stated.

Cite

arXiv 2609.28603 — Learning to Discover Interesting Mathematics, submitted 2026-09-23. HuggingFace Daily Papers, 2026-09-28, 11 upvotes — a popularity signal from that community and not a quality or importance ranking (source). Author list and affiliations: 2 WebSearch passes, not read first-party.