alternative RRR method the vector $x$ in also includes the node values, pre- and post-activation, for some number of instantiations of the network. More significantly, the flow field for RRR, $$F(x)=P_B(2 P_A(x)-x) - P_A(x),$$ is not the gradient of any function. Fixed points of the flow, defined by $F(x^*)=0$, are a problem for SGD training because optimization ceases without the guarantee that the loss is zero. For RRR this same condition implies $$P_B(2 P_A(x^*)-x^*) = P_A(x^*)=x_\mathrm{sol},$$ or a solution to because the $x_\mathrm{sol}$ so defined lies in both $A$ and $B$. The relationship between $x^*$ and $x_\mathrm{sol}$ is illustrated in Figure 1 for sets $A$ and $B$ that locally are flats in the ambient space (red and green lines), a model that applies to our use of the algorithm. In a feasible instance (left panel) with unique solution $A\cap B=\{x_\mathrm{sol}\}$, the orbit of $x$ converges to any point $x^*$ in the space orthogonal to $A$ and $B$ at $x_\mathrm{sol}$ (black dashed line). All of these fixed points are associated to the same solution point, $x_\mathrm{sol}$. When the instance is infeasible (right panel) the orbit converges to the same space it did in the feasible instance except that now it also 