Nested-transport depth consistency

1. Integration point: replace a standard recurrent or equilibrium block by an under-relaxed unroll. Given input u, initialize h_0, compute h_{t+1}=(1-alpha)h_t+alpha F_theta(h_t,u), and retain hidden states h_0,...,h_n. During training, choose a short depth m and long depth n>m; the task head uses h_n, while the new regularizer couples all short and long states. Use alpha >= 0.5. 2. Pseudocode: compute h[0]=init(u); for t=0,...,n-1, set h[t+1]=(1-alpha)h[t]+alpha F_theta(h[t],u). Construct path masses pi^t either from the paper's lattice-path dynamic program using the displayed per-path weight alpha(alpha^2 ell)^k(1-alpha)^(n+m-1-j-2k), or initially use normalized empirical masses obtained by counting sampled paths. Build z from equation (18), then compute L_NTDC=sum_{i,j} z[i,j] ||stopgrad(h[i])-h[j]||^2. Backpropagate L_task+lambda L_NTDC. 3. Mathematics versus estimation: equation (18) and the transport loss are exact once pi^m and pi^n are known. Estimate ell cheaply with one-step Jacobian-vector power iteration, ell approximately ||J_F(h)v||_2/||v||_2, and renormalize the resulting path masses if the estimated contraction is above one. For a first MVP, bypass exact pi derivation and compare the paper's nested z against uniform all-pairs weights and diagonal-only matching. 4. First experiment: train a 4-layer iterative MLP or small GRU on sequential MNIST and CIFAR-10 classification, with m=4,n=8, against the same model without the regularizer and standard deep supervision. Measure long-depth accuracy, short-depth accuracy, hidden-state variance, and the number of iterations required for the prediction to agree with the long-depth prediction. Success is higher accuracy at depth m, reduced variance across stopping depths, or at least 25% fewer inference iterations at unchanged accuracy.