Faithful Latent Fixed-Point Solver

1. Integration point: choose an expensive iterative component with state v∈R^n, such as an implicit layer, recurrent solver, optimization inner loop, or equilibrium model whose baseline update is v_{k+1}=S(v_k). Add an encoder E_θ:R^n→R^m and decoder D_φ:R^m→R^n with m≪n, and parameterize a latent update T_ψ:R^m→R^m. During inference, initialize z_0=E_θ(v_0), iterate z_{k+1}=T_ψ(z_k), and return D_φ(z_k) instead of running S for many steps. 2. Pseudocode: sample states v from baseline trajectories; compute v_next=S(v); set z=E_θ(v) and z_next=T_ψ(z); accumulate L=||E_θ(v_next)−z_next||² + λ||D_φ(z_next)−v_next||²; sample latent states z̃ and add β||D_φ(T_ψ(z̃))−S(D_φ(z̃))||², plus reconstruction loss ||D_φ(E_θ(v))−v||². Optimize θ, φ, and ψ jointly while S remains a fixed teacher. 3. Computed from the paper's mathematics: the two commutation residuals, latent rollout, and decoded fixed-point test. Estimated empirically: the stable neighborhood, contraction factor ρ from the slope of log||v_{k+1}−v_k||, and the smallest useful latent dimension m. Reject models whose decoded fixed points have large ||S(D_φ(z*))−D_φ(z*)||. 4. First cheap experiment: use a 10-dimensional synthetic contractive fixed-point map and a small deep-equilibrium MLP whose baseline requires 30–100 solver iterations. Compare the uncompressed solver, Anderson acceleration, and the faithful latent solver at equal output residual. Success means at least 2× fewer evaluations of S, lower decoded fixed-point residual, and no increase in divergence frequency under perturbed initial states.