Generic and Robust Schmidt Structure of Weighted Phase Networks: Matching Laws, Parameter Geometry, Higher-Order Obstructions, and Gauge-Centered Stability

This work develops a fixed-bipartition theory of Schmidt-rank capacity, parameter accessibility, higher-order algebraic obstruction, and quantitative Schmidt-spectrum stability for diagonal weighted phase networks acting on full-support product states. For a bipartition V=P Q, the output Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise sector, if G_× is the crossing interaction graph and ν(G_×) is its maximum matching number, the generic Schmidt rank obeys the exact law SR_PQᵍᵉⁿ = 2ν(G_×). The upper bound follows from a minimum vertex cover, while a maximum matching provides a nonvanishing symbolic minor. The corresponding uniform-d qudit model gives the generic law dν(G_×). The paper then separates interaction support from parameter geometry. For the uniform one-parameter family on the complete bipartite graph Kp,q, the exact rank is rank K_θ = min\ p+1,\, q+1,\, ord(eiθ) \, with ord(z)=∞ when z is not a root of unity. More generally, linear parameter embeddings are analyzed through signed assignment spectra. In particular, for a fixed template with one run-time parameter, θₑ(t) = t Wₑ, almost every sufficiently nondegenerate template W reaches the full matching capacity 2ν(G_×) for almost every t. Thus one scalar run-time control can generically access the full fixed-cut Schmidt-rank capacity even though the prepared coupling template may contain edge-dependent information. For rank-one templates W=uvT, the generic rank is determined exactly by subset-sum diversity: rgen = min\ Nᵤ,\, Nᵥ \, where Nᵤ and Nᵥ are the numbers of distinct subset sums generated by the components of u and v. This interpolates between low-rank highly symmetric tied families and one-parameter families with maximal exponential Schmidt rank. Higher-order weighted hypergraph phases admit an exact union factorization K = U C(λ) VT, so that the Schmidt-rank problem reduces to the symbolic rank of the union coefficient matrix C(λ). With at most two crossing hyperedges, symbolic generic rank equals structural matching rank. A three-hyperedge example on a 23 cut gives the minimal obstruction rstructural=4, rsymbolic=3, showing that higher-order union algebra can create intrinsic parameter dependencies invisible to support matching. Exact finite enumeration verifies the corresponding small-system minimality statement within the simple crossing-hypergraph model. The second part of the work develops a quantitative robustness theory. Full support alone is shown to be an algebraic rank condition rather than a quantitative coherence condition: no strictly positive state-independent entanglement floor follows from full support. Using a matching-generated reference state, the exact matching-core Schmidt spectrum is obtained. Residual phases are then analyzed modulo arbitrary cut-local diagonal rephasings. The nonlinear finite-phase quotient distance is distinguished from its additive Hoeffding/ANOVA tangent representative. After removing the cut-local phase directions, the quotient covariance factorizes as ΣR(h,g) = CovP( XLₕ, XLg ) CovQ( XRₕ, XRg ). This yields the Schmidt-spectrum stability bound s - s(M) ₂² ≤ εT ΣR ε, together with periodic lift optimization over the phase torus. For simple pairwise residual networks, the quotient covariance is diagonal. For balanced product inputs, ΣR = 1/16I, and hence s - s(M) ₂ ≤ 1/4 ε ₂. By contrast, higher-order residual interactions retain an intrinsic two-sided overlap geometry. For general full-support product inputs, ΣR(h,g) ≠ 0 if and only if the two hyperedges overlap on both sides of the bipartition: Lₕ ∩ Lg ≠ and Rₕ ∩ Rg ≠ . The resulting framework separates four structural layers: Matching geometry controls generic Schmidt-rank capacity; Template geometry controls accessibility and parameter degeneracy; Higher-order union geometry controls structural-versus-symbolic rank obstruction; Cut-local quotient geometry controls quantitative Schmidt-spectrum robustness. The accompanying reproducibility bundle contains the LaTeX source, exact and deterministic verification scripts, audit and revision records, verifier output, and SHA256 manifest. The computational checks include exact small-hypergraph enumeration, symbolic-rank verification, linear-embedding tests, quotient-covariance identities, two-sided overlap support checks, and deterministic Schmidt-spectrum stress tests.