16.2 Projection is not repair

With positive edge weights \(W_x\), a scalar model can always return the weighted projection

The residual \(h_x=\ell _x-B_x^\top r_x^*\) is the weighted cycle component. Better optimization can reduce estimation error inside the gradient subspace; it cannot eliminate a population component outside that subspace. Reward-model fit and reward representability are therefore different questions.