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Compatibility entropy and the spectrum of a Thorp sweep

OpenAI · September 26, 2026

Lean

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Abstract

We prove that the Thorp shuffle on $n=2^d$ cards mixes in $\Theta(d)$ physical shuffles. After a sufficiently large fixed number of coordinate sweeps, the total-variation distance of the full permutation from uniform tends to zero as $d\to\infty$.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope notes for family 238 (lean/docs/238.md, 2026). States that the formalization covers the weighted compatibility theorem for balanced A×D rectangles with an exp(C₀n^{54/100}) loss (ThorpWeightedCompatibility.lean); the bounded regular-moment and other moment/rank conclusions are not included. [family-level]

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Digests and expositions

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Cite

@misc{OAI:Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026,
  author = {{OpenAI}},
  title = {{Compatibility entropy and the spectrum of a Thorp sweep}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026/paper.pdf}{OAI:Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-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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