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Signed tensor densities and diagram budgets for the Thorp shuffle

OpenAI · September 26, 2026

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Abstract

We prove that a fixed number of coordinate sweeps of the Thorp shuffle on $2^d$ cards makes the squared L2 distance of the full permutation density from uniform tend to zero as $d\to\infty$. In particular, the total-variation mixing time is $\Theta(d)$ physical shuffles.

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Cite

@misc{OAI:Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026,
  author = {{OpenAI}},
  title = {{Signed tensor densities and diagram budgets for the Thorp shuffle}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026/paper.pdf}{OAI:Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/238.md. The Lean statements have not been checked by people against the paper's claims.

Family 238: Optimal logarithmic mixing of the Thorp shuffle

Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\Theta(\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

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