Ky-Fan certificate for tensorized layers

1. Integration point: replace a dense linear map in a small MLP or transformer projection with a tensorized map W = Σ_{l=1}^m A_l ⊗ B_l. For an input reshaped into compatible tensor dimensions, apply each Kronecker factor without explicitly materializing W, or materialize W only for the baseline comparison. Add a regularizer to the training loss that penalizes selected Ky Fan certificates U_k, especially U_1 for Lipschitz control and U_r for the top-r energy. 2. Pseudocode: for each layer: for l in 1..m: sA[l] = svdvals(A[l]); sB[l] = svdvals(B[l]); c = sort_desc(sum_l kron(sA[l], sB[l])); U = c.cumsum(0); R = alpha * sum(k in K, weight[k] * (U[k] / target[k])^2); loss = task_loss + R; optimizer.step(). The symbol c is the paper's majorization right-hand side and U[k] is its certified bound on the sum of the largest k singular values of the assembled layer. 3. Mathematical computation versus estimation: compute factor SVDs exactly when factors are small; for larger factors use 5-10 steps of randomized subspace iteration and treat the resulting singular values as approximate factor spectra. The sorting and cumulative sums are exact for the estimated spectra. Do not assume equality: periodically materialize W and compute its true top-k singular values to measure certificate looseness. 4. First experiment: train a 2-layer CIFAR-10 MLP with a 1024-to-1024 hidden projection represented by m=4 Kronecker summands, comparing no spectral regularization, ordinary dense spectral-norm penalty, and the proposed K={1,8,32} certificate penalty at matched parameter count and FLOPs. Track validation accuracy, gradient norm variance, true spectral/Ky Fan norms of W, and training stability under a 2x larger learning rate. Success is lower true top-k norms and fewer exploding-loss runs at equal accuracy, or equal accuracy with smaller factor ranks or parameter count; also report the ratio U_k / ||W||_(k) to quantify whether the theorem's bound is useful rather than vacuous.