Optimal mixing of the Thorp shuffle
Optimal mixing of the Thorp shuffle is the subject of a family of twelve manuscripts dated 26 September 2026 in OpenAI's public mathematics collection (github.com/openai/math). The Thorp shuffle cuts a deck into two halves and interleaves them, the order within each interleaved pair being chosen by a fair coin. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original results
- Claim. The Thorp shuffle on N = 2d labelled cards mixes in Θ(log N) complete shuffles: after 1600d shuffles the law of the whole permutation converges to uniform in total variation as d → ∞, uniformly over the initial deck, while a support count gives a lower bound of 2d − O(1). Source: Optimal-order mixing of the Thorp shuffle, with eleven supporting manuscripts listed in the collection's index.
- A supporting inequality. One of the supporting manuscripts, A strict four-row permanent inequality and permutation moments, concerns a strict form of a four-row permanent inequality.
- Formalization, as described by the repository. The scope note lean/docs/238.md states that the Lean formalization proves the optimal-order mixing statement (after 1600d shuffles, together with the support lower bound) and a number of supporting estimates, with some asymptotic conclusions of individual supporting papers outside the selected statements.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 14 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | J. D. Jones, JD-Jones-ASES/four-row (GitHub; Palomar PALOMAR-2026-10-08-000007, version 1) | Extension (formal) | A Lean formalization of sharp robustness and stability refinements of the four-row permanent inequalities of Carlen–Lieb–Loss and Bristiel–Caputo: the exact total-variation radius 1/24, the optimal variance-deficit coefficient (1 − 24r)/9, the nonnegative equality cases and sharp endpoint entropy constants. The record notes that its explicit interior exponent is sufficient rather than optimal. | Registered in the Palomar registry on 8 October 2026. |
| 2026-10-12 | Issue #1 in the GitHub repository punk01-shaka/hadwiger-nelson-audit | Critique (formalization setup) | The issue states that several of OpenAI's Comparator configurations declare fully written definitions as definition_names, so that Comparator checks only their names and types; it lists this family's OccupiedOverlap and SpinAngle among them and demonstrates accepted fake solutions for three other results. |
Open issue; no fake solution is demonstrated for this family. |
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.