Cactus-Graph Phase Budgeting

Implement a graph module containing serial chains and parallel skip branches, such as a residual-in-residual MLP or a transformer layer with multiple attention paths. Associate every edge with a scalar phase parameter theta_k and a positive gain parameter; these can be learned or used only as certificate variables. During a periodic certification step, estimate each edge Jacobian on a minibatch using Jacobian-vector products, approximate its SRG by a polar grid of complex points, multiply point sets along serial paths, and add the resulting sets across parallel paths. Compute d=min_z|1+z|. If exact point-set operations are expensive, use interval boxes in log-gain and phase coordinates. Add the hinge penalty or use feedback gain control: multiply branch output by c_q, with c_q updated to c_q min(1,d/d_min), whenever the margin is too small. The chord and arc composition rules and -1 avoidance condition are imported from the paper; finite Jacobian probes and set approximations are estimated. First test on CIFAR-10 using a 12-layer residual CNN or 24-layer residual MLP, comparing unrestricted parallel branches against phase-budgeted branches at equal parameter count. Track training loss, gradient norms, Jacobian conditioning, and d versus depth. The falsifiable prediction is a transition when the aggregate enclosure touches -1: the unbudgeted model should show oscillatory optimization and a sharp increase in ||(I+sum_q P_q)^(-1)||, whereas gain control should keep d above d_min and prevent the transition.