Clustering as Structural Order Parameter and the Layered Admissibility Architecture of Finite Relational Systems

We establish that the clustering coefficient acts as the primary structural order parameter governing phase behavior in finite weighted relational systems under damped wave dynamics, and identify a universal ordering of admissibility layer failure that holds across structurally diverse graph families. Three partially independent admissibility layers are identified—diffusion-geometric, bounded propagation coherence, and post-geometric remnant—each sensitive to the same order parameter but exhibiting distinct failure signatures, distinct collapse timescales, and distinct post-collapse structure. Multi-seed simulation across system sizes N ∈ {100, 250, 400} and five independent random seeds demonstrates that the 1 clustering coefficient C correlates with all primary behavioral metrics at Spearman ρ > 0.96 and p < 10−25 across 45 independent observations per condition in Watts-Strogatz graphs. Strong correlations are confirmed in Erd˝os-R´enyi graphs (ρ ≈ 0.87–0.90); ER graphs at reasonable mean degree probe only the low-tointermediate clustering regime (C ≲ 0.50). Barab´asi-Albert graphs reveal that structurally heterogeneous systems require family-specific variables—hub clustering (ρ = −0.923) for post-collapse isolation and degree heterogeneity (ρ = 0.915) for the dual-mode stability boundary— establishing structural homogeneity as a background condition for single-parameter order parameter descriptions. Across all three graph families and 1,575 independent observations, causal admissibility failure consistently precedes spectral admissibility collapse at rate 1.000. Two critical clustering thresholds are located empirically: C ∗ ≈ 0.15, below which horizon-like overload trapping is unreliable, and C ≈ 0.40, below which post-geometric dynamic persistence is extinguished under strong overdamping at large N. A dual-mode stability boundary γ ∗ (C) is identified separating wave-causal from diffusion-dominant propagation regimes; in the finite-crossing regime it follows γ ∗ (C) = A exp(−αC), confirmed at R2 = 0.847–0.924 across all tested system sizes, and scales with system size as γ ∗ (C, N) ∼ N −δ with δ ∈ [0.78, 1.17]. The scoring robustness of the primary results is established in the companion paper [6]: the two-class transport structure and layer ordering are preserved across 35 scoring variants. Results are interpreted without appeal to physical ontology; physical correspondence is deferred to Papers 7–9.