Right but in reality IIUC w ∈ R^d and it's x = w / ||w|| ∈ S^(d-1) and then given r = x - Qmse^-1( Qmse( x ) ) the scalar you use is derived as ||r|| (I'm missing a couple subscript twos there I think).

I was primarily aiming to confirm my understanding given the author's omission but also the scalar is subtly different than in your linked explanation (although conceptually equivalent).