Spectral-Gated Parallel Best Responses

First, integrate the mechanism as an inner optimizer for a network whose trainable parameters are divided into blocks, for example four LoRA adapters attached to the same transformer layer, or four experts trained on partially conflicting routing losses. During each outer minibatch, freeze activations and form a block-local quadratic model from the current gradient and a cheap curvature approximation. For block $i$, use a diagonal, Kronecker-factored, or low-rank approximation $H_{ii}$; estimate cross-block products $H_{ij}v$ with Hessian-vector products instead of constructing the full Hessian. Pseudocode: compute each block gradient $g_i$ and curvature blocks; set $D=\operatorname{blkdiag}(H_{ii})$ and implement $E=H-D$ as a Hessian-vector-product operator; estimate $\widehat{\rho}=\rho(D^{-1}E)$ using 5 to 20 power iterations; if $\widehat{\rho}<0.8$, execute simultaneous updates $z_i\leftarrow[D^{-1}(r-Ez)]_i$; if $0.8\leq\widehat{\rho}<1$, use damping with $\alpha$ selected from $\{0.1,0.25,0.5,0.75\}$; if $\widehat{\rho}\geq1$, execute blocks sequentially or increase within-block regularization via $H_{ii}\leftarrow H_{ii}+\lambda I$. The exact paper mechanism is the Jacobi stability condition; neural-network curvature and spectral radius are empirical estimates, so log both $\widehat{\rho}$ and the actual quadratic residual norm $\|Hz-r\|$. The first cheap experiment should use a two-layer MLP on MNIST with two parameter blocks and a tunable cross-block penalty $\gamma$ in $L=L_1+L_2+\gamma\langle z_1,z_2\rangle$. Compare Adam, parallel best response, damped best response, and sequential best response. The falsifiable prediction is a sharp divergence or oscillation transition when $\widehat{\rho}(D^{-1}E)$ crosses one: below one, inner residuals decay geometrically with slope approximately $\log\widehat{\rho}$; above one, simultaneous updates fail while sequential updates remain stable. The measured stability boundary should agree with the spectral prediction within approximately 20 percent.