Symplectic quadratic interaction layer

Integrate this as a replacement for the first dense pairwise-interaction block in a small MLP or graph neural network. Let the incoming tensor be \(z\in\mathbb{R}^{B\times d}\), with \(d=2g\) channels grouped into \(g\) pairs. First compute a linear term \(u=W_1z\). For each pair \((a_i,b_i)\), compute the symplectic interaction \(s_i=z_{a_i}z_{b_i}\); optionally compute cross-pair terms only through tied coefficients determined by the fixed matrix \(J\), rather than learning an independent coefficient for every ordered pair. Concatenate \([1,z,s]\), apply normalization and a projection \(W_2\), and add the result residually to the original block. A practical pseudocode version is: z = input; u = W1 @ z; s_i = z[:,a_i]*z[:,b_i] for i=1..g; phi = concat(ones(B,1), z, s); out = W2 @ phi + bias; y = z + gate*tanh(out). For a stronger version, binarize or softly binarize channels with \(p=\operatorname{sigmoid}(z/\tau)\), evaluate sampled quadratic forms \(q_r\) using \(q_r(x+y)=q_r(x)+q_r(y)+x^\top Jy\pmod 2\), and add the displayed \(\mathcal{L}_{\mathrm{symp}}\) term. Generate \(U\) from cheap symplectic basis operations: swap two coordinate pairs, swap members within a pair with sign adjustment, and apply elementary shears; these satisfy \(U^\top JU=J\). The paper supplies the exact cocycle and dimension count; the real-valued relaxation, gate, temperature \(\tau\), and sampled transformation distribution are engineering choices. The first experiment should use a two-layer MLP on CIFAR-10 embeddings or a molecular graph network, with \(d=16\) or \(32\), comparing a dense quadratic layer, a standard MLP, and this tied layer at equal hidden width. Measure validation accuracy, parameter count, and training loss per FLOP. A positive result is equal-or-better accuracy with at least 25% fewer interaction parameters, or faster convergence under random symplectic channel permutations.