Graded residual geometry for degenerate inverse networks

Integrate this at the output residual and inverse-solver loss, not inside every hidden layer initially. Use a small differentiable map F_theta, an inverse model G_phi, or an unrolled optimizer that seeks x satisfying F_theta(x) approximately equal to y. First choose a block partition of the output; for a synthetic problem, define the Y_i blocks explicitly, while for a real model use groups of coordinates, channels, or task heads. Estimate each block order before training. At a reference input x_bar, sample random directions d_k and scales t_k in {10^-1, 10^-2, 10^-3}. Compute a_ik = ||P_i(F(x_bar + t_k d_k) - F(x_bar))||_2 and fit the slope of log(a_ik) against log(t_k). Assign block i to the nearest stable integer slope, retaining only slopes with adequate signal-to-noise; otherwise assign i = 1. During training, compute r_i = P_i(F_theta(G_phi(y)) - y), evaluate the smooth log-sum-exp surrogate above, and backpropagate it jointly with a small ordinary MSE term, for example L = L_grade + 0.1 times the sum of ||r_i||_2^2. An unrolled solver can additionally accept an update only when q_p(r_new) <= q_p(r_old), or can enforce per-block targets ||r_i|| <= rho^i. The mathematical quantities computed directly are the projections, fractional powers, and graded gauge; the constant M, valid neighborhood, and order labels are estimated empirically. Start with F(x_1, x_2) = (x_1, x_2^2) near x_2 = 0, then test a rank-deficient MLP autoencoder or nonlinear inverse problem. Compare ordinary MSE, manually weighted MSE, and the graded loss at equal parameter count and FLOPs. Success means lower second-block error at the same training budget, fewer failed inverse solves, and a tighter empirical relation between reconstruction displacement and q_p(y - y_bar).