Auxiliary-energy neural optimizer

Integrate this at the optimizer-update stage, after the minibatch gradient is computed and before parameters are written back. Start with an MLP on MNIST or CIFAR-10, where $\theta$ is the flattened parameter vector conceptually but all operations should be implemented layerwise. Choose $H=\lambda I$ or a diagonal running curvature estimate, $P=\operatorname{diag}(p_i)$ with $p_i=1/(\sqrt{v_i}+\epsilon)$, and $E(\theta)=\mathcal{L}_{\mathrm{batch}}(\theta)$ after shifting it by a nonnegative running baseline if necessary. At step $n$, evaluate the minibatch loss $E_n$ and gradient $g_n$, set $b_n=g_n/\sqrt{E_n+C}$, and initialize $r_n=\sqrt{E_n+C}$ on the first step. Solve the scalar/vector coupled system without forming a dense matrix: substitute $\theta_{n+1}=\theta_n-\tau P(H\theta_{n+1}+r_{n+1}b_n)$, giving $\theta_{n+1}=A_n^{-1}(\theta_n-\tau P r_{n+1}b_n)$ with $A_n=I+\tau PH$. Compute two diagonal or conjugate-gradient solves, one for $A_n^{-1}\theta_n$ and one for $A_n^{-1}P b_n$, then determine the scalar $r_{n+1}$ from the auxiliary equation. Finally recompute the minibatch loss and optionally damp the step if the actual loss increases sharply. The paper supplies the auxiliary-variable normalization and energy-law structure; $H$, $P$, $C$, and minibatch estimates are engineering choices. First compare against AdamW and SGD on a 2-layer 512-unit MLP at equal forward/backward FLOPs, sweeping learning rates above the AdamW stability threshold. Measure loss explosions, gradient norms, final accuracy, and the modified-energy sequence; success is a substantially larger stable learning-rate range and smoother loss descent, with no worse accuracy at matched compute.