Exact Gaussian randomized range finder

Input a fixed matrix $A\in F^{m\times n}$, a positive integer sketch width $\mathrm{\ell }$, and a Gaussian draw primitive as in Standard real and normalized complex Gaussian matrix laws. Draw $\mathrm{\Omega }\in F^{n\times \mathrm{\ell }}$ with iid entries in the chosen field and form $Y=A\mathrm{\Omega }$. Select $Q$ deterministically by scanning columns of $Y$ in order, subtracting projections onto already retained columns, skipping zero residuals, and normalizing each nonzero residual. In the complex case fix each phase by making its first nonzero coordinate positive real. Return this orthonormal basis $Q$ and the approximation $\widehat{A}=Q{Q}^{\ast }A$.

Empty ranges return the empty basis. All operations use exact arithmetic. No truncated rank-$k$ factorization, finite-precision stability result, or deterministic error certificate is part of the procedure.