The scaled Householder QR implementation is backward stable

Embed each active orthogonal representative into the full row dimension and each local perturbation by zeros outside its active block. All active row counts are at most $m$, so [F1] bounds every local error by ${\kappa }_m$ times the current active-block norm, hence by that times the current full-matrix norm. A completed full step therefore has norm at most $1+{\kappa }_m$ times the preceding norm. Induction bounds the matrix before step $k$ by $(1+{\kappa }_m{)}^k\mathrm{\parallel }A{\mathrm{\parallel }}_F$.

Transport each local error back through the preceding orthogonal factors. Orthogonal multiplication preserves Frobenius norm, as follows by summing the preserved squared column norms. The transported errors sum to $\mathrm{\Delta }A$, and exact column overwrites make the final lower block zero. Thus [F2] identifies $A+\mathrm{\Delta }A=\widehat{Q}\widehat{R}$, with $\mathrm{\parallel }\mathrm{\Delta }A{\mathrm{\parallel }}_F\le {\kappa }_m{\sum }_{k=0}^{n-1}(1+{\kappa }_m{)}^k\mathrm{\parallel }A{\mathrm{\parallel }}_F={\beta }_Q\mathrm{\parallel }A{\mathrm{\parallel }}_F$.

For the separate vector, [F1] bounds each update discrepancy by ${\kappa }_m$ times its current norm. The identical norm induction and orthogonal transport argument bound its total discrepancy from the exact product by ${\kappa }_m{\sum }_{k=0}^{n-1}(1+{\kappa }_m{)}^k\mathrm{\parallel }b{\mathrm{\parallel }}_2={\beta }_Q\mathrm{\parallel }b{\mathrm{\parallel }}_2$. Since ${\kappa }_m=O(u)$ for fixed $m$, the claimed fixed-dimension backward and application bounds follow.