OpenAI · 238-05 · 2026-09-26 · Probability and statistical mechanics
Conditional permutations in a revealed switching environment
preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026
Family 238. Optimal logarithmic mixing of the Thorp shuffle.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that the Thorp shuffle randomizes labeled cards in 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.
Fix half the labels in a Thorp shuffle on positions and reveal their complete trajectories. We prove that, after an explicit absolute number of coordinate sweeps, the conditional permutation of the remaining labels approaches uniform in expected total variation as , uniformly in the initial layout. The resulting constructions give full-deck mixing in a constant multiple of d physical shuffles.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 238: Optimal logarithmic mixing of the Thorp shuffle. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.