Generic Schmidt Rank of Weighted Phase Networks: Matching Laws, Parameter Tying, and Higher-Order Union Obstructions

This work studies bipartite Schmidt rank in continuously weighted diagonal phase networks, with particular emphasis on the distinction between interaction support, parameter freedom, and higher-order algebraic dependencies. For a fixed bipartition and a full-support product input, the Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise qubit sector, the main result gives an exact generic matching law: rgen = 2ν(G_×), where ν(G_×) is the maximum matching number of the crossing bipartite interaction graph. The symbolic rank, maximum attainable rank, and almost-everywhere physical rank coincide. The paper then shows that the interaction support alone does not determine the generic Schmidt rank once the parameter family is constrained. For a uniformly weighted complete bipartite network Kp,q, the exact rank is rank K_θ = min\ p+1,\, q+1,\, ord(eiθ) \, with infinite multiplicative order interpreted in the natural way. The K2,2 example therefore exhibits a rank stratification (4 → 3 → 2) when passing from independently weighted phases to a uniform one-parameter family and then to the Clifford point. For genuine higher-order phase interactions, the cross-cut kernel is factorized through a union-coefficient matrix. This reveals a different obstruction: even with independent hyperedge weights, repeated algebraic use of the same phase parameters can force the symbolic rank below the structural matching rank. The paper proves that no such obstruction occurs with at most two crossing hyperedges and gives an explicit three-hyperedge example with rₛₜᵣ=4, rsym=3. Exact finite enumeration further certifies, within the stated simple crossing-hypergraph model, that five total cut vertices are the smallest size at which this structural-versus-symbolic rank mismatch occurs. The results distinguish three mechanisms controlling cross-cut Schmidt-rank capacity: pairwise interaction combinatorics, external parameter tying, and intrinsic higher-order union-algebraic dependencies. The manuscript is accompanied by exact verification code and reproducibility checks for the finite classification statements. Version: v0.2r3