Algebraic-Invariant Residual Layer

Integrate the construction into a residual MLP or recurrent/state-space block. Start with the scalar rational feature y=1/(1+x^2), represented by G(x,y)=(1+x^2)y-1=0; apply the constraint independently to each feature coordinate, making B diagonal and inexpensive. Store both x and y. For each block, compute v=dt*MLP(concat(x,y)); update x_new=x+v. Evaluate A=partial_x G(x_new,y) and B=partial_y G(x_new,y) analytically or with autodiff. Compute the tangent correction c=B_damped_dagger*A*v, where B_damped_dagger=(B^T B+lambda I)^(-1)B^T and lambda is initially 1e-6. Set y_new=y-c. Optionally perform one or two Newton corrections y_new <- y_new-(partial_y G)^dagger G(x_new,y_new). Pseudocode is: initialize y=H(x); repeat v=dt*F_theta(x,y), x=x+v, y=y-Bdagger*A*v, then repeat y=y-solve(B,G) if projection is enabled. The tangent correction is the paper-derived component; Newton projection is the practical discrete safeguard. Log ||G(x,y)||, activation magnitudes, gradient norms, loss, and wall-clock overhead. First test a 24-, 64-, and 128-layer residual MLP on CIFAR-10 and a synthetic task requiring repeated composition of the rational map. Compare ordinary rational activations, unconstrained auxiliary updates, projection-only updates, and the full method at equal width and optimizer settings. Success requires substantially smaller constraint residuals, fewer unstable runs, and lower loss or higher accuracy at depth 64 or greater without a large throughput penalty.