Certified Cusp Scanner for Implicit Layers

(1) Integration point: use an implicit layer or equilibrium recurrent block defined by z*=z*+g_theta(z*, input, p), equivalently f_theta(z,p)=z-g_theta(z,input,p)=0. Run the scanner during validation, after training, or every K optimizer steps on representative inputs. The parameters p should be two scalar controls such as residual gain and normalization strength; they may be trainable scalars or two-dimensional projections of a larger parameter vector. (2) Pseudocode: choose a converged equilibrium zbar with Newton or fixed-point iteration; compute J=J_f(zbar,p) using autodiff; initialize v as the smallest-singular-vector approximation of J and w as the corresponding left singular vector; assemble F_NN=[f; Jv; J^T w; v^Tv-1; w^Tw-1; w^Tv]; form DF_NN by forward/reverse autodiff; compute an approximate inverse A with LU or damped least squares; take x_next=x-AF_NN(x) with backtracking, and repeat until the augmented residual is small. (3) Compute from the paper's mathematics the augmented zero-finding problem, inverse defect, and derivative-variation estimate. Estimate the Lipschitz bound empirically by sampling xbar+delta directions and measuring ||A(DF(x)-DF(xbar))||/||delta||; estimate the local spectral gap with a few Arnoldi or power iterations. Reject a candidate if the residual, inverse defect, or Lipschitz product is too large. (4) First experiment: use a 2-layer tanh DEQ on MNIST or CIFAR-10 with residual gain p1 and damping p2, comparing ordinary training against a penalty triggered when the scanner finds a nearby fold or cusp. Evaluate convergence failures, equilibrium solver iterations, validation loss, Jacobian spectral abscissa, and classification accuracy. Success means fewer failed equilibrium solves and lower validation loss at equal compute; deliberately retained cusp regions should produce reproducible two-state behavior rather than uncontrolled divergence.