Monotone Resolvent Elimination Layer

Integrate this as an implicit residual/MLP block with a retained feature vector x in R^{d_*} and an auxiliary vector u in R^{d_**}. Parameterize one joint piecewise-affine map R(x,u)=[R_*(x,u),R_**(x,u)] using linear layers and ReLU or leaky-ReLU branches. Apply spectral normalization or explicit matrix rescaling to the joint linear operators so that the estimated global Lipschitz constant of R is at most one. In the forward pass, initialize u^0=0, or reuse the previous u for temporally related inputs, and perform K iterations: compute (v_*,v_)=R(x,u^k), then set u^{k+1}=(1-alpha)u^k+alpha v_. Return y=v_* from the final iteration, optionally followed by a learned linear projection and residual skip x+y. Backpropagate through the K iterations for the first implementation; implicit differentiation can be added once fixed-point convergence is reliable. The paper supplies the reduced-operator definition, the resolvent nonexpansiveness target, and the piecewise-affine structure. Estimate the auxiliary contraction factor empirically with finite differences or power iteration: q=max ||R_(x,u+delta)-R_(x,u)||/||delta|| on held-out minibatches. Use alpha=1 if q<0.9; otherwise use alpha=min(1,0.9/(q+1e-6)). Track the fixed-point residual ||u^{k+1}-u^k||, activation norms, gradient norms, and perturbation amplification. First test on a four-layer MNIST or CIFAR-10 MLP against a standard residual MLP with matched parameters and FLOPs. The desired signal is fewer unstable training runs, bounded input perturbation amplification, and equal or better accuracy at a learning rate where the baseline becomes unstable. Also report iteration overhead and test whether smaller auxiliary dimensions reduce activation memory without hurting accuracy.