Matroid-circuit equivariant message passing

Integrate this as a replacement for a hypergraph-convolution or graph-attention block in a graph or set neural network. Inputs are element features X with shape n x d and a fixed list of matroid circuits C_1,...,C_m. First build the sparse incidence matrix A with A[i,k]=1 when element i belongs to circuit C_k. Compute circuit features with a permutation-invariant reducer, for example h_k = MLP_orbit(k)(mean_{i in C_k} X[i]); the orbit label orbit(k) is obtained by applying a matroid automorphism tool offline and grouping circuits into orbits. Scatter circuit features back to elements using M_i = sum_{k:A[i,k]=1} W_orbit(k) h_k / sqrt(|C_k|), then update X_new = X + MLP([X,M]). Use only orbit-tied parameters; do not give each circuit an independent embedding. For a stronger operator-inspired version, alternate two channel maps corresponding to the paper's sigma_x sigma_z ordering, such as W_x(W_z(h_k)), while retaining shared orbit parameters. During training, sample a small automorphism P from the precomputed automorphism group and add L_eq above; with exact parameter tying, this loss should be numerically near zero and mainly detect implementation bugs. The mathematics determines incidence normalization, orbit tying, and the equivariance target; MLP widths, nonlinearities, and optimizer settings are empirical. The first experiment should use synthetic classification on randomly relabeled copies of small uniform and graphic matroids, comparing an untied hypergraph MLP, ordinary hypergraph message passing, and this layer at equal parameter count. Measure accuracy under unseen relabelings, examples needed to reach a fixed accuracy, and runtime versus circuit count. A positive signal is near-perfect relabeling consistency and higher test accuracy or lower sample requirement than the untied baseline; on real graph benchmarks, compare against a standard hypergraph neural network with matched FLOPs.