Total variation cutoff for Kac's walk on the sphere

We prove cutoff in total variation distance for the discrete-time Kac walk on $S^{n-1}$ started from a coordinate vector. The cutoff occurs at $
C_{\mathrm{BRW}}n\log n$, where $C_{\mathrm{BRW}} \approx 3.8916$ is an explicit constant determined by the speed of the leftmost particle in a branching random walk. In particular, the cutoff location is not at the conjectured time $ 2n\log n$.