Sketched curvature-subspace optimizer

Integrate this into a block curvature optimizer between gradient computation and the parameter update. Let the flattened parameter dimension be n, choose block size k such as 8 or 16, sketch size s such as 4k to 8k, and a fixed sparse signed sketch Omega implemented as a hash-and-sign map. Generate candidate directions from the current gradient, momentum direction, and additional Hessian-vector products on random probes or retained directions. Maintain both each full direction and its sketch. Pseudocode: initialize Q and Z as empty; for each candidate v, compute z = Omega v, c = Z transpose times z, replace v by v minus Qc and z by z minus Zc, reject the direction when the sketched residual norm is below a threshold, otherwise append v divided by its sketched norm to Q and z divided by its norm to Z. Compute W_j = A v_j using automatic-differentiation Hessian-vector products or Fisher-vector products. Form H_ij = v_i transpose times W_j and G_ij = v_i transpose times v_j, symmetrize both matrices, and solve H Y = G Y Theta. Set U = VY and use the selected modes in the damped update Delta = -U diag(1/(theta_i + lambda)) U transpose g. Increase lambda until the step norm is at most rho. Compute the sketch operations from the paper directly; estimate embedding quality by testing the ratio ||Omega Vx|| divided by ||Vx|| for random x. First test on a two-layer MLP or ResNet-18 on CIFAR-10 against SGD, AdamW, and the same block optimizer with full Gram-Schmidt, matching Hessian-vector-product counts. Record wall-clock optimizer overhead, loss decrease per second, validation accuracy, condition number of G, and generalized-eigensolver failures. A positive result is at least 10 percent lower orthogonalization overhead with unchanged accuracy, or faster loss descent at equal wall-clock time.