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

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.

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.