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Random coordinate frames and partial permutation laws

OpenAI · September 26, 2026

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Abstract

For the Thorp shuffle on $2^d$ cards, we prove that the worst-start total-variation distance after $32800d$ physical shuffles tends to zero as $d\to\infty$. Combining our fixed-list estimate with the companion Fourier transfer improves this bound to $512d$ shuffles. These results follow from bounds on partially observed permutation laws in random coordinate frames.

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Cite

@misc{OAI:Random-coordinate-frames-and-partial-permutation-laws-September-26-2026,
  author = {{OpenAI}},
  title = {{Random coordinate frames and partial permutation laws}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Random-coordinate-frames-and-partial-permutation-laws-September-26-2026/main.pdf}{OAI:Random-coordinate-frames-and-partial-permutation-laws-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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