Transition Front Dynamics and Transport Universality Classes at the Dual-Mode Stability Boundary in Finite Relational Systems

We investigate the propagation front velocity at the dual-mode stability boundary γ*(C) identified in Paper 5 of this series, where wave-causal and diffusion-dominant propagation regimes coexist at comparable strength in finite weighted relational systems. Within the Watts-Strogatz graph family, the front velocity at γ*(C) is approximately independent of both clustering coefficient C and system size N, with mean v* = 1.524 ± 0.188 (CV = 0.123) across 19 crossing detections spanning N ∈ {100, 250, 400} and C ∈ [0.19, 0.60]. Cross-family normalization reveals that the quantity κ = v* × √N/λ_max identifies two candidate transport universality classes. Structurally homogeneous families (Watts-Strogatz and Erdős-Rényi) converge to κ ≈ 2.20 — with WS giving 2.195 ± 0.628 and ER giving 2.208 ± 0.200 (n=47), a cross-family difference of 0.6%. Structurally heterogeneous Barabási-Albert graphs yield a distinct κ ≈ 1.00 (0.993 ± 0.203, n=89), consistent with localization-like transport renormalization in hub-dominated networks. Within BA, κ increases monotonically with the attachment parameter m as hub dominance decreases, approaching the Class 1 value at large m. The primary predictor of the BA transport class correction is the spectral ratio λ₁/λ_max — the balance between global propagation coherence and local spectral concentration — which also emerges as the governing structural variable for the γ*(C) transition boundary in Paper 5. This convergence suggests that the dual-mode transition in finite relational systems is governed by a single spectral-structural balance parameter that simultaneously determines when the transition occurs and how propagation behaves at the transition. The general transport renormalization framework v* = κ_ref × (λ_max/√N) × Φ(L) is proposed, where κ_ref ≈ 2.20 is the Class 1 reference value and Φ(L) is a localization correction functional that approaches unity for homogeneous systems and takes smaller values as hub concentration increases. Physical interpretation — including structural analogy to the null surface in relativistic spacetime — is presented with appropriate qualification.