Residual-score decomposition from square-root Lasso

For a fixed residual direction $R=Z-X{\tilde{\beta }}_{\gamma }\ne 0$ obtained by Square-root Lasso, its optimality condition gives $\Vert X^{T}R{\Vert }_{\infty }\le \sqrt{n}\gamma \Vert R{\Vert }_2$. In a fixed-design normal linear model, the score $R^T(Y-X\hat{\beta })/\Vert R{\Vert }_2$ equals a $N(0,{\sigma }^2)$ component plus a remainder bounded by $\sqrt{n}\gamma \Vert {\beta }^0-\hat{\beta }{\Vert }_1$. The direction must be fixed or independent of the response noise for that normality claim.