AI Assisted
Mathematical Proofs

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

The preprint is unrefereed and unformalized, relying on author review and rewriting of the AI-discovered argument.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

Reported in OpenAI's math repository with a machine-checked Lean formalization of the main theorem. The repository's formal artifacts are self-assessed and have not been independently audited or rebuilt for this catalogue.

The preprint is unrefereed by journal reviewers. The authors provide a Lean 4 formalization of the main theorem in an associated repository, but the formal build has not been independently reproduced for this catalogue.

The preprint is unrefereed and unformalized, with no independent check recorded.

Only the HorizonMath certificate for 3.6961 is recorded here. What is contested is that certificate, not the inequality; whether R(k,k) ≤ 3.6961^(k+o(k)) holds is not decided by it. Other recent bounds on this constant are not catalogued in this entry.

The improvement applies to the two-probe upper bound in the full-table model. The preprint is unrefereed and unformalized.

The preprint presents an explicit 14-vertex counterexample. It is an unrefereed preprint without a formal theorem prover artifact, though the finite search and graph properties are directly testable.

The paper proves a reduction to the three-branch-point case rather than proving the full conjecture unconditionally. The preprint is unrefereed and unformalized.

The preprint provides an explicit 8-point counterexample in seven dimensions. The paper has not undergone formal peer review or computer verification in an interactive theorem prover, though the finite arithmetic is directly checkable.

The preprint is unrefereed, unformalized, and relies on author calculation and verification.

The counterexample is an explicit continuous piecewise quadratic function on a semialgebraic partition of Euclidean space. The preprint is unrefereed and unformalized, with verification based on author checks and symbolic validation.

This very recent preprint is unrefereed and has not yet received broad specialist review. Its smooth, unforced result is distinct from earlier lower-regularity blowup constructions and from the concurrent forced-Euler work of Alpöge and Buckmaster. The public Lean project reports a full formalization, but its review status is self-assessed and the build has not been reproduced for this catalogue.

This is a same-day release, not yet peer reviewed or broadly accepted by specialists. The construction uses smooth external forcing, exactly as allowed by alternatives C and D of the official Millennium Prize formulation; it does not establish unforced blowup. OpenAI does not intend to claim the prize. The public Lean project reports full, sorry-free coverage, but its review status is self-assessed and the build has not been reproduced for this catalogue.

This is an AI-produced formalization of existing mathematics, not a new proof strategy or a new theorem. It builds on the Imperial College FLT project, flt-regular and Mathlib, and reproducing the independent checks requires substantial compute.

The compared Lean theorem uses at least (k-1)n/2 + 1 edges. When (k-1)n is odd this is one edge stronger than the sharp classical hypothesis, so it is a marginally weaker result than the full conjecture.

Lean checks a counterexample to the matrix formulation. The classical implication from that formulation to Köthe's original statement is described but is not part of the formal development, and no independent ring-theory review is recorded.

The four-page preprint is unrefereed, has no formal artifact, and records author verification but no independent check.

The result is relative to a quantum oracle and does not separate QMA(2) from QMA in the unrelativized setting. The preprint is not formally verified or independently reviewed.

This is a preliminary, unrefereed preprint without formalization or independent review.

The Lean artifact reportedly covers existence of counterexamples at every base size, but not all additional base-two families or diameter computations; no third party audited statement-to-paper fidelity.

The counterexamples do not settle the corresponding question for the Landau equation. The preprint is unrefereed, unformalized and lacks an independent check.

The preprint is disputed. A public technical objection alleges an inconsistent recurrence factor that invalidates the zero count used for the degree bound; the only whole-proof verification claimed in the paper is another language-model reading.

The field matters: the 47-multiplication result is over F₂; the later characteristic-zero construction is a different claim. Neither establishes optimality.

The preprint is unrefereed and unformalized, with no independent check recorded. The authors supplied the known structural constraints and wrote the proofs and manuscript.

The result settles this natural function, not the general Boyar–Find question. The Lean statements have not received a recorded third-party semantic audit.

The claimed Lean development survives at a pinned commit rather than the repository's main branch. It proves a gluing conjecture reported to imply the percolation result; statement fidelity and the implication have not been independently audited here.

The short preprint is unrefereed, unformalized and has no independent check; the AI disclosure does not assign individual proof steps.

The preprint is unrefereed, unformalized and has no independent check recorded.

The preprint is unrefereed, unformalized and has no independent check recorded.

This is an improvement to the record bound associated with Erdős

This is an unreviewed claim in a problem with many failed proofs. The manuscript explicitly conflicts with a published corrigendum, contains no AI disclosure itself, and the AI attribution is only from a public author statement.

The unrefereed preprint has no formalization or independent check, and the disclosure credits the AI agent with the initial counterexample rather than every part of the final proof.

This settles only the (4,6) case, with an enormous explicit threshold. The preprint is unrefereed, unformalized and has no independent check.

This is a formalization of an existing theorem, not a new prime-gap bound. The public flagship Lean theorem assumes Bombieri–Vinogradov as a hypothesis because that theorem is not yet in Mathlib.

The preprint is unrefereed, unformalized and has no independent check recorded.

Formal coverage is reported as partial, and the bundled audits are not independent replication.

The preprint is unrefereed and unformalized. The authors checked and reworked the proof, but no independent review is recorded.

Verification details are reported in the sources; proof artifacts have not been rebuilt for this catalogue.

A positive proportion on the critical line is not a proof of the Riemann hypothesis.

Existing literature already contains a solution.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

This episode is not a disproof of Collatz. A kernel implementation bug invalidated the certificate; checker versions matter.

The original notes report a novelty and attribution dispute, rather than an allegation that the proof is wrong. See the linked coverage.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

The original notes report a novelty and attribution dispute, rather than an allegation that the proof is wrong. See the linked coverage.

An independent audit found a polarity error in a printed greedy-conditioning lemma and supplied a local repair. The correction does not independently verify the later sampleability, alignment, rounding or main parallel-repetition arguments; the Lean artifact was not rebuilt here.

Reported by OpenAI; peer review and local proof rebuild are not recorded.

The authors report checking the arguments and references themselves. This entry does not assert external peer review or machine verification.

Verification details are reported in the sources; proof artifacts have not been rebuilt for this catalogue.

The notes report a higher-dimensional counterexample; the two-dimensional question is separate.

The notes describe conditional Lean developments with unformalized classical inputs, not unconditional certificates.

Verification details are reported in the sources; proof artifacts have not been rebuilt for this catalogue.

Verification details are reported in the sources; proof artifacts have not been rebuilt for this catalogue.

This is a source-reported AI reproof following a human disproof, not a claim of first discovery. The domain is the real numbers. The reported seven successes out of eight trials were not independently graded here.

The paper covers slack δ ≥ 1; the linked Lean coverage is reported as δ = 1 only. Do not label the whole theorem Lean-verified without an audit.

Lean coverage is reported; no local rebuild was performed.

The notes report native_decide use in this artifact.

Lean coverage is reported; no local rebuild was performed.

Lean coverage is reported; no local rebuild was performed.

The notes report native_decide use in this artifact.

Lean coverage is reported; no local rebuild was performed. Rundström reportedly found an independent solution concurrently.

The notes report native_decide use in this artifact.

Lean coverage is reported; no local rebuild was performed.

Human reading only; no formalization is listed.

Human reading only; no formalization is listed.

The linked Lean development is reported to check a seed identity, not the complete theorem.

The reported bound is Var(|X|²) ≤ 8n under the stated isotropy and log-concavity assumptions. AI worked from human-supplied prompts and mathematical context. No formal artifact is listed.

Human reading only; no formalization is listed.

Human reading only; no formalization is listed.

Verification details are reported in the sources; proof artifacts have not been rebuilt for this catalogue.

Human reading only; no formalization is listed.

Public formal code does not by itself establish that its statements match the original competition problems.

Disclose the dataset version and access conditions; neither a score change nor data access alone proves training contamination.

An independent human proof is reported a day earlier; track discovery priority separately from formal verification.

The notes report formal coverage of the upper bound; the lower-bound construction is a separate obligation.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

A formal variant may be weaker than the full numbered question. No local rebuild was performed.

A formal variant may be weaker than the full numbered question. No local rebuild was performed.

A formal variant may be weaker than the full numbered question. No local rebuild was performed.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

The sequence may contain several conjectures; this entry refers only to the statement in the linked artifact. Its correspondence to the OEIS text has not been audited.

Human authors developed the proofs; the AI contribution was a method suggestion.