Minor-preserving complex network layer

Replace a dense complex projection or feed-forward matrix with a product of $L$ sparse factors. Let each $T_k$ be unit-diagonal lower or upper triangular with nonnegative off-diagonal magnitudes, let each $E_k$ be positive diagonal, and attach independent phases $e^{i\phi}$ to the trainable edges: $w_e=r_e e^{i\phi_e}$ with $r_e\geq0$. The positive companion uses the same magnitudes and sets all phases to zero. In the forward pass compute z=x; z0=abs(x); for k in 1..L: z=Tk_complex(Ek_complex(z)); z0=abs(Tk_complex)@(abs(Ek_complex)@z0). On each batch, sample two to eight row-column pairs $(I,J)$ with sizes $r=1$ and $r=2$, compute the corresponding determinants using torch.linalg.det, and add $\lambda_{minor}\mathcal{L}_{minor}$ only when the complex minor is below a chosen fraction of its positive companion. For $r=1$, this is an entrywise constraint; for $r=2$, it detects pairwise rank collapse. The paper-derived quantities are the path-family determinant expansion and the inequality $|\Delta_{I,J}(A)|\leq\Delta_{I,J}(A_0)$; subset sampling, margins, and penalty weights are empirical. First test on a 4-layer complex MLP or complex attention value projection on CIFAR-10, comparing dense complex, real dense, and the structured factorization at matched parameter count. Track accuracy, Jacobian condition numbers, gradient variance, and throughput. Success means equal or better accuracy with fewer near-singular Jacobians and stable training at larger learning rates.