Multivariate Structural Curriculum

Integrate this into synthetic graph-data generation or MoE/router curriculum construction. Choose z to mark total graph nodes or computation blocks and u to mark one controllable feature: branching nodes, skip connections, active experts, or long-range edges. Build a recursive grammar whose constructors add powers of z and optionally u; for example, a leaf contributes z, unary composition contributes zA, binary merge contributes zuA^2, and an unmarked merge contributes zA^2. Store a two-dimensional table f[n,k] by polynomial convolution, truncating at maximum size N and attribute count K. To sample a batch with exact target (n,k), backtrack through constructor choices with probability equal to each constructor's number of valid child decompositions divided by f[n,k]. Use the diagonal construction operationally as choosing one canonical subtree and reusing or mirrored it in a coupled branch, thereby producing tied or invariant structures instead of independently sampled branches. Begin training with low k, then increase the target distribution over k, or mix coefficient slices uniformly so frequent easy structures do not dominate. The coefficient table and sampling probabilities are mathematical; task difficulty, augmentation semantics, and the schedule are empirical. Test on CIFAR-10 or a synthetic graph-classification benchmark with a six-layer GNN, comparing this curriculum against iid random graphs matched for node count. Track validation accuracy on held-out high-branching graphs, calibration across k, and learning curves at equal examples. Success is improved extrapolation to rare structural regimes without degrading performance on ordinary graphs.