Exact Mirror Geometry of ReLU Policy Gradient Reparameterization in Homogeneously Constrained LQR

We study the policy-gradient dynamics induced by a neural reparameterization in scalar homogeneously constrained LQR. Its optimal policy is a two-slope controller fixed by two feedback gains, and it can be realized by a bias-free residual ReLU network arbitrarily many weights. Because the parameterization map is non-injective, the controller and weight problems share values and minimizers but not gradient dynamics; \textit{we characterize the difference exactly.} First, we show every stabilizing controller satisfying the KKT conditions of the constrained gain problem is its global minimizer, with no convexity or gradient-dominance assumption. Second, we show gradient flow on the weights is exactly a projected mirror flow on the gains: it stays feasible and stabilizing, converges globally, is independent of the network width, and depends on the initialization only through two scalar masses that interpolate between a Euclidean and a non-Euclidean regime. Finally, we find a Bregman discretization lifts back to the weights exactly, converges, and identifies the active set in finitely many steps.