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

Abstract

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 $V=P\sqcup Q$, the output Schmidt rank is reduced exactly to the complex rank of a cross-cut phase kernel. In the independently weighted pairwise sector, if $G_\times$ is the crossing interaction graph and $\nu(G_\times)$ is its maximum matching number, the generic Schmidt rank obeys the exact law $$\operatorname{SR}_{P\vert{}Q}^{\mathrm{gen}} = 2^{\nu(G_\times)}.$$ 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 $d^{\nu(G_\times)}$. The paper then separates interaction support from parameter geometry. For the uniform one-parameter family on the complete bipartite graph $K_{p,q}$, the exact rank is $$\operatorname{rank} K_\theta = \min\left\{ p+1,\, q+1,\, \operatorname{ord}(e^{i\theta}) \right\},$$ with $\operatorname{ord}(z)=\infty$ 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, $$\theta_e(t) = t W_e,$$ almost every sufficiently nondegenerate template $W$ reaches the full matching capacity $2^{\nu(G_\times)}$ 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 $W=uv^{\mathsf T}$, the generic rank is determined exactly by subset-sum diversity: $$r_{\mathrm{gen}} = \min\left\{ N_u,\, N_v \right\},$$ where $N_u$ and $N_v$ 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 $$K = U C(\lambda) V^{\mathsf T},$$ so that the Schmidt-rank problem reduces to the symbolic rank of the union coefficient matrix $C(\lambda)$. With at most two crossing hyperedges, symbolic generic rank equals structural matching rank. A three-hyperedge example on a $2\vert{}3$ cut gives the minimal obstruction $$r_{\mathrm{structural}}=4, \qquad r_{\mathrm{symbolic}}=3,$$ 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 $$\widehat{\Sigma}_R(h,g) = \operatorname{Cov}_P\left( X_{L_h}, X_{L_g} \right) \operatorname{Cov}_Q\left( X_{R_h}, X_{R_g} \right).$$ This yields the Schmidt-spectrum stability bound $$\left\vert{} \mathbf{s} - \mathbf{s}^{(M)} \right\vert{}_2^2 \le \boldsymbol{\epsilon}^{\mathsf T} \widehat{\Sigma}_R \boldsymbol{\epsilon},$$ together with periodic lift optimization over the phase torus. For simple pairwise residual networks, the quotient covariance is diagonal. For balanced product inputs, $$\widehat{\Sigma}_R = \frac{1}{16}I,$$ and hence $$\left\vert{} \mathbf{s} - \mathbf{s}^{(M)} \right\vert{}_2 \le \frac{1}{4} \left\vert{} \boldsymbol{\epsilon} \right\vert{}_2.$$ By contrast, higher-order residual interactions retain an intrinsic two-sided overlap geometry. For general full-support product inputs, $$\widehat{\Sigma}_R(h,g) \neq 0$$ if and only if the two hyperedges overlap on both sides of the bipartition: $$L_h \cap L_g \neq \varnothing \quad \text{and} \quad R_h \cap R_g \neq \varnothing.$$ 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.