The OpenAI release

372 result families published by OpenAI on 6 October 2026, each scored for its likely effect on applied mathematics and algorithms and for the effort the problem had absorbed.

The collection holds 722 manuscripts written by an unreleased OpenAI model. Its README reports about 4,000 problems posed and about three hours of compute per result. 127 families have a main theorem in the Lean formalization catalogue, 108 have Lean proofs of supporting statements only, and 137 have none.

Applied score

The applied score rates the route from the theorem to a computation, an algorithm or a model. It takes each result as stated and says nothing about whether the proof is correct.

• 5. Usable as stated today: an inequality to quote, a guarantee for a method already in use, or a condition checked by hand.
• 4. Settles a question practitioners face, with a short path to methods or a clear change in where effort goes.
• 3. A sharp tool or limit with an indirect path, including algorithms that are polynomial on paper and out of reach in code.
• 2. A path exists only through further theory, or the result confirms what practice already assumes.
• 1. No route to computation, algorithms or modelling is visible today.

A score of 1 describes today. The Ledger lists fields that sat at that point for a century before an application arrived.

Usable today

Three results score 5. Each can be used as stated, with no new algorithm to implement.

• A function of a matrix is at most twice its size on the numerical range (No. 325, functional analysis).
• Edge swapping reaches a uniformly random network for every degree sequence (No. 131, theoretical computer science).
• A reaction network in which every reaction lies on a loop never loses a species (No. 149, dynamical systems and ergodic theory).

Several results that read as faster algorithms score 3 or lower. The manuscripts on contingency tables, edit distance and log-concave sampling state that their bounds are asymptotic or are not practical running times, and almost-linear matching is built on almost-linear maximum flow.

Difficulty score

The difficulty score rates how long the problem stood and how much work it drew. The year in the Posed column is approximate and is blank where the origin of the problem is diffuse.

• 5. A famous problem, typically open fifty years or more, or the central conjecture of its field.
• 4. A well-known open problem with decades of partial results.
• 3. A recognised conjecture with its own literature.
• 2. A recent or specialist question.
• 1. A new question with little prior work.

Applied score against difficulty

Theoretical computer science accounts for 4 of the 8 results scored 4 or 5. Of the 105 problems at difficulty 5, 87 have an applied score of 1 or 2.

The results

Click a column header to sort and a row to open the catalogue description, the scoring note and the manuscripts. Descriptions are OpenAI's, reproduced under the Apache 2.0 licence of the collection. Lean: main means a main theorem is in the formalization catalogue, part means supporting statements only.

The Votes column records readers' views of how important each result is. One vote per result.