LAR active correlation invariant ($X_A^{\mathsf{T}}(Y-X\hat{\beta }(\lambda ))/n=\lambda s_A$)

Each least angle regression segment starts with all active scores at magnitude $t$. Moving with direction $d_A=G_{AA}^{-1}{s}_A$ changes the score vector to $t{s}_A-(t-\lambda )G_{AA}{d}_A=\lambda s_A$. The first-hit rule bounds every inactive score in magnitude by $\lambda$. This is the key Lasso optimality link.