Path-Coupled Neural Stability Margin

1. Integration point: use an implicit MLP or recurrent equilibrium block $z=F_\theta(z,x,\rho,u)$, or approximate the same construction around an ordinary MLP by treating the hidden representation as $z$ and defining the residual $R(z)=z-F_\theta(z,x,\rho,u)$. Let $\rho$ scale an adverse transformation such as feature masking, additive noise, blur, label corruption, or distribution shift. Let $u$ be a small learned control vector that gates channels, shifts normalization statistics, or activates a rank-4 adapter. 2. Pseudocode: initialize $\rho=0$, $u=0$, and $z$ from the clean forward pass; at each step propose several $\Delta\rho$ values and candidate controls $u$; solve the residual equation with 3--10 fixed-point or Newton iterations; estimate the smallest singular value of $J=\partial R/\partial z$ using 5--10 Lanczos or power iterations; accept the candidate maximizing $\Delta\rho-\lambda_c\|u\|_2^2-\lambda_s[\max(0,\varepsilon-\sigma_{\min}(J))]^2$; then backpropagate through accepted steps and add $-\alpha M$ to the training objective. 3. The continuation update and cumulative margin are taken from the paper's path-coupled construction. Jacobian singular values, control costs, and failure thresholds are estimated numerically. 4. First cheap experiment: train a 3-layer MLP on MNIST with Gaussian corruption severity as $\rho$, compare standard noise training against the coupled method, and use a 2-dimensional feature gate as $u$. Sweep continuation step size and record accuracy, $\sigma_{\min}(J)$, and cumulative margin. The prediction is that the fixed-path baseline reaches $\sigma_{\min}(J)=\varepsilon$ at a smaller $\rho$ than the controlled model, while the controlled trajectory increases measured margin $M$ monotonically with control budget. A falsifiable target is at least a 10% increase in collapse severity at equal clean accuracy, with the empirical marginal stability cost $\Delta\rho/\Delta\|u\|^2$ changing sharply near the uncontrolled failure point.