Generic Schmidt-Rank Width of Fixed Weighted Phase Networks: Maximum-Matching Width, Template Diversity, and Single-Parameter Global Saturation

This work studies the global multipartite entanglement width generated by fixed pairwise weighted-phase networks acting on full-support product inputs. It develops the global-width consequences of the fixed-cut weighted-phase theory established in the preceding study, Schmidt-Rank Capacity and Spectral Robustness in Fixed Weighted Phase Networks; the corresponding public research record is available at DOI 10.5281/zenodo.21858429. For a graph $G=(V,E)$ with a fixed real edge-weight template W=(Wₑ)e∈ E, we consider the commuting weighted-phase evolution UW(t) = exp( i t ∑_e=,v\∈ E Wₑ nᵤ nᵥ ), with a single scalar evolution parameter t. For every bipartition AAᶜ, let νG(A) be the maximum matching number of the crossing graph G[A,Aᶜ]. The main result globalizes the fixed-cut matching law. There exists an open, dense, full-Lebesgue-measure class of fixed templates W such that, outside one locally finite exceptional set of evolution times, the same scalar t simultaneously saturates the matching capacity of every nontrivial bipartition: log₂ SR_AAᶜ(ΨW(t)) = νG(A) for all nontrivial cuts AAᶜ. Consequently, χwd(ΨW(t)) = mmw(G) for generic fixed weighted-phase networks, where χwd is Schmidt-rank width and mmw(G) is the maximum matching width of the interaction graph. This provides a continuous weighted counterpart to the standard graph-state relation between Schmidt-rank width and rank-width. For complete support Kₙ, mmw(Kₙ) = /3, so almost every nonuniform fixed template attains χwd = /3, which is the largest Schmidt-rank width permitted for any n-qubit pure state. Uniform complete-graph templates behave very differently. Their exact width is χwdᵘⁿⁱᶠ = log₂ min\ /3+1,\, ord(eiθ) \, with ord(eiθ)=∞ allowed. Thus the same complete interaction support and the same one-dimensional run-time control exhibit a sharp template hierarchy: Clifford χwd=1, generic uniform χwd=Θ(log n), generic nonuniform χwd=Θ(n). This separation is not restricted to dense complete graphs. Since rw(G) ≤ mmw(G), graph families with linear rank-width transfer directly to fixed weighted-phase families with linear generic Schmidt-rank width. In particular, known deterministic bounded-degree high-rank-width constructions yield sparse $O(n)$-coupler networks with generic Schmidt-rank width Θ(n). The width separation has an exact tensor-network consequence. For generic uniform complete templates, an explicit subcubic tree tensor network exists with optimal maximal bond dimension Dunif = /3+1 = Θ(n), whereas generic nonuniform complete templates require Dgen = 2n/3 = 2Θ(n). Thus fixed template diversity, rather than run-time control dimension alone, can change exact global tree-tensor complexity from polynomial to exponential while the interaction support and the number of run-time control parameters remain unchanged. The computational interpretation is deliberately limited. Superlogarithmic Schmidt-rank width lies outside the logarithmic-width hypothesis underlying the standard low-width exact-TTN simulation guarantee, and linear width is a particularly strong realization of this separation. However, large or linear Schmidt-rank width is neither asserted to be necessary nor sufficient for universal measurement-based quantum computation, hard classical simulation in general, fault tolerance, or quantum advantage. Version v0.2r2 is the frozen theorem-core release. It preserves the mathematical results of v0.2r1 while finalizing the claim boundary concerning tensor-network simulation, measurement-based quantum computation, and computational advantage.