Certified contraction implicit layer

1. Integration point: implement an implicit replacement for an MLP residual block, transformer feed-forward block, or SSM transition. Given x, solve z=phi(Wz+Vx+b) by damped fixed-point iteration or Anderson acceleration, then return y=Uz+Rx. Parameterize W so its certified gain remains below one, for example W=(q/L)W_bar/(||W_bar||_2+epsilon), with q=sigmoid(alpha), or use spectral normalization after each optimizer step. 2. Pseudocode: compute W and estimate s=||W||_2; set q=Ls; reject or project parameters if q>=0.95; iterate z <- (1-gamma)z+gamma phi(Wz+Vx+b) until the residual is below tolerance; in backward mode form A_phi and solve (I-A_phi W)^T lambda=dJ/dz, then use the transpose implicit-gradient equations. 3. Computed from the paper: the fixed-point representation, contraction inequality, and implicit Jacobian. Estimated empirically: spectral norm, local convergence factor, solver iterations, and gradient condition number. 4. First cheap experiment: train a four-layer-equivalent implicit MLP on MNIST or CIFAR-10 with identical parameter count, comparing unconstrained DEQ-style training against q ceilings 0.5, 0.8, 0.95, and 1.05. Measure residual decay versus iteration k and sweep q across one. The prediction is geometric residual decay with slope log(q) for q<1, a sharp increase in solver iterations and gradient norm near q=1, and frequent nonconvergence for q>1; the measured asymptotic decay factor should be within approximately 20 percent of L||W||_2.