Rank-aware sketch-and-solve least squares

Input $A\in {\mathbb{K}}^{n\times p}$, $b\in {\mathbb{K}}^n$, a supplied sketch $S\in {\mathbb{K}}^{m\times n}$, and a finite nonnegative integer SVD sweep budget $N$, where $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$. Form $C=SA$ and $c=Sb$, and call Rank-thresholded least squares by the singular value decomposition on $(C,c)$ with threshold $\tau =0$ and budget $N$.

If the call returns exact-arithmetic convergence with an exact SVD record, return success, the exact rank $r=\mathrm{rank}(C)$, the canonical minimum-norm vector $\widehat{x}=C^{+}c$, and the separately computed sketched and physical residuals $\mathrm{\parallel }C\widehat{x}-c{\mathrm{\parallel }}_2$ and $\mathrm{\parallel }A\widehat{x}-b{\mathrm{\parallel }}_2$. The exact-arithmetic solve-error certificate is zero. If the SVD call returns inconclusive, return its status and supplied partial diagnostics without claiming a minimizer. In finite precision, return the computed candidate and the two residual reports separately; mark the rank unverified unless a certified interval separates every singular-value comparison from zero, and report a solve-error certificate only when the solver supplies one. No finite-precision accuracy guarantee is part of this procedure.