Minimum-motion curvature-targeted preconditioner

Integrate this mechanism at the optimizer-metric update, not in the network forward pass. Partition parameters into blocks of 256-2048 coordinates and initially use diagonal metrics, G_j=diag(exp(g_j)), because the affine-invariant distance then becomes an inexpensive squared Euclidean distance in log-metric coordinates. Maintain the current metric G_0 for each block and use the optimizer step theta <- theta - eta G_0^{-1} grad(theta). Every R training steps, estimate a block curvature H_hat using diagonal generalized Gauss-Newton or Fisher statistics; for small blocks, obtain lambda_max and lambda_min using 2-4 Hessian-vector products and a few Lanczos iterations. Set a planning horizon T=4 or 8 and initialize g_1,...,g_T from the current g_0. For 5-20 inner iterations, compute M_T=G_T^{-1/2}H_hatG_T^{-1/2}, estimate its extreme eigenvalues, form c=max(0,log(lambda_max+eps)-log(lambda_min+eps)-log K), and minimize beta*c^2+(alpha/(2T))*sum_j ||g_{j+1}-g_j||^2 by backpropagation through the planned metrics. Execute only g_1, then shift the plan and re-optimize at the next controller update. For full SPD blocks, parameterize G_j=G_{j-1}^{1/2}exp(V_j)G_{j-1}^{1/2}, with symmetric low-rank V_j, and use the matrix-log affine distance. The mathematical quantities are the generalized spectral target, affine-invariant distance, and kinetic coefficient; curvature estimates, K, alpha, beta, R, and T are empirical hyperparameters. First test on a 2-layer MLP or small ResNet on MNIST or CIFAR-10 against AdamW and a standard EMA diagonal preconditioner at equal optimizer-update budgets. Record loss versus steps, validation accuracy, gradient-norm variance, measured generalized condition number, metric jump size, and controller overhead. Success means faster loss descent or lower gradient variance at equal FLOPs without more than 10% wall-clock overhead; failure is no improvement in conditioning or convergence stability.