https://openai-aftermath.mathnt.net/index.php?action=history&feed=atom&title=Optimal_mixing_of_the_Thorp_shuffleOptimal mixing of the Thorp shuffle - Revision history2026-10-15T00:13:22ZRevision history for this page on the wikiMediaWiki 1.46.2https://openai-aftermath.mathnt.net/index.php?title=Optimal_mixing_of_the_Thorp_shuffle&diff=45&oldid=prevAftermath bot: New pages: Euclidean Ramsey classification (172), Thorp shuffle mixing (238), mass-action permanence (149), with Palomar and GitHub follow-ups (UC-0037); Main Page links2026-10-14T04:05:02Z<p>New pages: Euclidean Ramsey classification (172), Thorp shuffle mixing (238), mass-action permanence (149), with Palomar and GitHub follow-ups (UC-0037); Main Page links</p>
<p><b>New page</b></p><div>'''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 ([https://github.com/openai/math 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.<br />
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== The original results ==<br />
* '''Claim.''' The Thorp shuffle on ''N'' = 2<sup>''d''</sup> labelled cards mixes in Θ(log ''N'') complete shuffles: after 1600''d'' 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 2''d'' − O(1). Source: [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026 ''Optimal-order mixing of the Thorp shuffle''], with eleven supporting manuscripts listed in the collection's index.<br />
* '''A supporting inequality.''' One of the supporting manuscripts, [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026 ''A strict four-row permanent inequality and permutation moments''], concerns a strict form of a four-row permanent inequality.<br />
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/238.md lean/docs/238.md] states that the Lean formalization proves the optimal-order mixing statement (after 1600''d'' 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.<br />
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== Public follow-up work ==<br />
{| class="wikitable"<br />
! Date !! Work !! Relation !! Public claim !! Public status (checked 14 October 2026)<br />
|-<br />
| 2026-10-08 || J. D. Jones, [https://github.com/JD-Jones-ASES/four-row JD-Jones-ASES/four-row] (GitHub; Palomar [https://palomar-registry.org/entry?id=PALOMAR-2026-10-08-000007&version=1 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 − 24''r'')/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.<br />
|-<br />
| 2026-10-12 || [https://github.com/punk01-shaka/hadwiger-nelson-audit/issues/1 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 <code>definition_names</code>, so that Comparator checks only their names and types; it lists this family's <code>OccupiedOverlap</code> and <code>SpinAngle</code> among them and demonstrates accepted fake solutions for three other results. || Open issue; no fake solution is demonstrated for this family.<br />
|}<br />
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== Notes ==<br />
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.<br />
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