OpenAI · 238-04 · 2026-09-26 · Probability and statistical mechanics
Conditional information under deterministic coordinate sweeps
preprints/Conditional-information-under-deterministic-coordinate-sweeps-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.
We bound the information remaining after paths of specified cards in the Thorp shuffle on cards have been observed. After a fixed number of deterministic coordinate sweeps, the joint endpoint law of further cards is close to uniform on the available positions, on average over the observed paths, provided a fixed positive fraction of labels lies outside both lists. Combined with a Fourier transfer, these bounds give full-deck mixing after physical shuffles.
Preprint source and Lean scope
The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.
A Lean scope note is stored for family 238, but it does not name this preprint. It names 238-01, 238-03, 238-05, 238-06, 238-07, 238-08, 238-09, 238-10, 238-11, 238-12. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.