Optimal-order mixing of the Thorp shuffle on $2^d$ cards: is the mixing time $O(\log n)$?

Lineage and corrections

This event attributed to Unreleased internal OpenAI model

Companion: Compatibility entropy and the spectrum of a Thorp sweep evidence for this event

Companion: Routing densities and representation contraction for Thorp sweeps evidence for this event

Companion: From partial permutation information to Fourier bounds evidence for this event

Companion: Conditional coordinate sweeps and analytic transfer evidence for this event

Lean proof (OAI.ThorpResults.remaining_main) evidence for this event

Companion: Signed tensor densities and diagram budgets for the Thorp shuffle evidence for this event

Companion: Row-column symmetry and contraction of coordinate sweeps evidence for this event

Principal Theorem 1.1: for $n=2^d$, $\|q_d^{*1600d}-U_n\|_{TV}\to0$ as $d\to\infty$, uniformly over initial decks, and $\|q_d^{*t}-U_n\|_{TV}\ge1-2^{tn/2}/n!$, so $t_{mix}(d)\ge2d-O(1)$ and $t_{mix}(d)=\Theta(d)$. Companions give alternative proofs with other constants and the supporting Fourier and coordinate-sweep estimates. The constant 1600 is not claimed sharp; no cutoff or leading constant is determined, and the optimal-order question for general even deck sizes is explicitly not addressed. parent of this event

Optimal-order mixing of the Thorp shuffle on $2^d$ cards: is the mixing time $O(\log n)$? evidence for this event

VibeMathed record: Optimal-order mixing of the Thorp shuffle on $2^d$ cards: is the mixing time $O(\log n)$? evidence for this event

Companion: Conditional permutations in a revealed switching environment evidence for this event

Optimal-order mixing of the Thorp shuffle on $2^d$ cards: is the mixing time $O(\log n)$? parent of this event

Companion: Random coordinate frames and partial permutation laws evidence for this event

Companion: Conditional information under deterministic coordinate sweeps evidence for this event

Companion: Random-subspace tests and trace smoothing for coordinate sweeps evidence for this event