oai-math-revised beta

Browse › Probability and statistical mechanics › 238 Optimal logarithmic mixing of the Thorp shuffle

Conditional coordinate sweeps and analytic transfer

OpenAI · September 26, 2026

Lean

Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion

Edit NOTES.md · Existing issues · Source on GitHub

Abstract

For coordinate sweeps with near-uniform line laws and line sizes among powers of two in a suitable fixed interval, we prove a Schatten-moment bound after conditioning on any feasible collection of prescribed card trajectories. An analytic transfer to binary sweeps then shows that a fixed number of sweeps mixes the Thorp shuffle on $2^d$ cards in $O(d)$ physical steps, matching the order of the support lower bound.

If the PDF does not display here, open it directly: Open PDF.

Notes

No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.

Automatically collected notes (unreviewed)

work in progress

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 binary-sweep power contraction D^{-g} and the resulting mixing (BinarySweep.lean), plus the conditional moment estimate with its feasibility premise (CoordinateSweeps.lean); physical-shuffle law identification is not included. [family-level]

Possibly relevant literature (found by the agent, not checked)

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026,
  author = {{OpenAI}},
  title = {{Conditional coordinate sweeps and analytic transfer}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026/main.pdf}{OAI:Conditional-coordinate-sweeps-and-analytic-transfer-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.

Other papers in this family: