Green-Margin Residual Dynamics

1. Exact integration point: use a depth-indexed residual network or linear state-space model. Set each block to $z_{k+1}=A_kz_k+F_k(z_k)$, where $A_k$ is initialized as a diagonal or block-diagonal matrix with selected contracting and expanding channels, and $F_k$ is a small MLP. Estimate a discrete dichotomy projection $P_k$ from the known backbone; for a diagonal backbone, $P_k$ selects channels whose cumulative products contract forward. 2. Pseudocode: compute or refresh transition products $T_{k,j}$; form $G_{k,j}=T_{k,j}P_j$ for $j\le k$ and $G_{k,j}=-T_{k,j}(I-P_j)$ for $j>k$; estimate $L_j$ by one or two power iterations on Jacobian-vector products; compute $q_N=\max_k\sum_j\|G_{k,j}\|_2L_j$. Penalize $\max(0,q_N-q_0)^2$ with target $q_0=0.8$, or rescale $F_j\leftarrow q_0F_j/q_N$ when $q_N>q_0$. 3. Computed from the mechanism: the Green kernel and contraction inequality. Estimated empirically: Jacobian norms, projections for non-diagonal $A_k$, and finite-depth $q_N$. 4. First cheap experiment: train a 20- or 50-layer residual MLP on MNIST or CIFAR-10, comparing ordinary residual blocks with Green-margin blocks at matched parameter count. Sweep a global residual multiplier $\eta$ and record exploding activations, gradient norms, and test accuracy. The quantitative prediction is a sharp degradation boundary near $\eta_c=1/q_N(\eta=1)$: stable-gradient runs should occur for $\eta q_N<1$, while instability and rapidly increasing sensitivity should appear as $\eta q_N\to1$; the measured boundary should agree with the estimated threshold within roughly 20%. Also test the predicted sensitivity scaling $\|\delta z_N\|/\|\delta z_0\|\propto(1-q_N)^{-1}$ below the boundary.