The Mathocalypse: OpenAI Solves 372 Longstanding Mathematical Problems

OpenAI's Mathematical Breakthroughs

OpenAI has released 372 major mathematical results, including a proof of the Unique Games Conjecture (UGC), solving problems that have remained open for decades. This event, termed the "Mathocalypse," represents a significant shift in mathematical discovery, where AI models can now solve complex, longstanding open problems that previously required years of human effort.

Key Mathematical Results

Among the 372 results, several high-profile conjectures and problems in theoretical computer science and mathematical physics were solved:

Unique Games Conjecture (UGC): A central statement in complexity theory implying that many optimization problems are NP-hard even for approximations slightly better than semidefinite programming relaxation.
L=BPL: The proof that probabilistic logspace and deterministic logspace are equivalent, solving a major derandomization conjecture.
Computational Complexity: Breaking the O(n log n) barrier for the Fourier Transform and integer multiplication, achieving a running time of approximately O(n log 0.9999999999999 n).
Unitary Synthesis Problem: A positive solution proving that for every n-qubit unitary transformation U, there exists a classical oracle A such that U can be implemented in quantum polynomial time with access to A.
Quantum Complexity: A proof that Parity is not in QAC0 and a nearly 4th-power separation between randomized and quantum query complexity for total Boolean functions.
Matrix Multiplication: A new approach achieving matrix multiplication in O(n 9/4) time.
Computability Theory: A proof of the uncomputability of solving polynomial equations over the rational numbers, addressing a major open problem in computability theory.
Millennium Problems: Partial progress toward the Riemann hypothesis, the Hodge Conjecture, and the Birch-Swinnerton-Dyer Conjecture.

The "Alien" Nature of AI Proofs

While many of these results are accompanied by Lean certificates for mechanical verification, the natural language proofs are described as nearly impenetrable to humans. Complexity theorist Dana Moshkovitz notes that the proofs feel "written by someone who’s on psychedelics," citing a bizarre recursive construction and a completely new code with a noise test that does not resemble existing human mathematical techniques.

Basically the paper is so horribly written that it’s impossible to read it without AI help

This creates a new dynamic where humans must now use AI to help them digest and explain the AI-generated proofs, effectively turning the mathematician's role from a primary discoverer to an interpreter of "alien" logic.

Comparison of AI Communication Models

Two distinct approaches to communicating AI mathematical breakthroughs have emerged:

The OpenAI Model: Releasing raw, undigested proofs and Lean certificates to the world, triggering a race among humans to understand and explain the messy AI output.
The Anthropic Model: Partnering with human mathematicians (e.g., Virginia Williams and Josh Alman) to produce a digested, human-readable version of a result (such as the 3SUM and All-Pairs Shortest Paths problems) in exchange for compensation.

Model Capabilities and Efficiency

The results were produced by OpenAI's latest internal model, rather than a bespoke system of agents. On average, the model spent approximately 3 hours of compute per problem. The success rate was approximately 5%, as the model solved 372 out of 8,000 attempted longstanding open problems after a single 3-hour attempt.

Critical Perspectives and Verification Risks

Despite the mechanical verification provided by Lean, some researchers express concern regarding the validity of the translation from natural language to formal language. A paper cited in the discussion suggests that the process of autoformalization—translating natural language proofs into Lean—may not be semantically faithful, potentially leading to a mismatch between the NL proof and the Lean verification.

Additionally, some critics argue that the 5% success rate indicates a limitation in the model's capabilities, while others suggest that the cost of solving a longstanding mathematical problem—estimated at roughly $810 to $8,100 per solution based on API pricing—is remarkably low compared to the value of the mathematical progress achieved.