This repository serves as a formal public release of a collection of mathematical manuscripts and proof artifacts generated by an advanced, unreleased internal OpenAI model. The project was initiated to evaluate the model's capabilities on open research problems, specifically after existing benchmarks showed signs of saturation.

Scope and Methodology

The collection comprises 722 manuscripts organized into 372 "families," which include primary results, companion arguments, and alternative proofs across various mathematical disciplines. The core methodology relied on an iterative, compute-intensive process:

• Compute Utilization: Each result required, on average, three hours of "thinking" compute.
• Problem Scope: The model was subjected to approximately 4,000 distinct mathematical problems to arrive at the current catalog.
• Formalization: The repository emphasizes the transition from natural language manuscripts to formal proofs using the Lean proof assistant. While not all results are currently formalized, the project maintains an ongoing commitment to formal verification.
• Exceptions: While the vast majority of results were generated via a standardized automated procedure, select high-complexity tasks—specifically the proof of the Hodge Conjecture for CM abelian varieties and the work on a zero-free region for the Riemann zeta function—involved more extensive, specialized workflows, with the latter receiving human editing for readability.

Key Mathematical Result Families

The repository provides reasoning summaries for high-significance topics, including:

• Analytic Number Theory: Investigation into the zero-free region for the Riemann zeta function, targeting the $\text{Re}(s) > \frac{11}{12}$ threshold, and the irrationality exponent of $\pi$.
• Probability and Statistical Mechanics: Derivations of the Mézard–Parisi formula for diluted spin glasses and the study of spontaneous magnetization in the quantum Heisenberg ferromagnet.
• Algebra and Combinatorics: Developments regarding Kaplansky's direct-finiteness conjecture in characteristic two and the isomorphism of free group factors.
• Complexity Theory: Results on ordinary NP-hardness at the basic semidefinite threshold and quasipolynomial bounds for arithmetic progressions.

Repository Architecture

The files are structured to facilitate reproducibility and ongoing academic rigor:

• preprints/: Contains the PDF documentation, source LaTeX files, and specific build/citation instructions for individual manuscripts.
• Lean Integration: The project includes a Lean library and a formalization catalog, mapping manuscripts to their corresponding machine-checked proofs.
• Dynamic Updates: The repository employs a versioning system to track corrections and revisions, ensuring that both current and historical versions remain accessible to the research community.

This initiative represents an attempt to bridge the gap between Large Language Model output and verified formal mathematics, providing a structured archive of complex automated mathematical reasoning.