Triangular solidification: local motif redundancy drives a sharp structural transition in complex networks

We show that the minimum per-edge triangle count τ(G) acts as a structural control parameter in complex networks. When τ(G) = 0, at least one edge belongs to no triangle and forms a structural bottleneck; when τ(G) ≥ 1, every edge is covered by at least one triangle and the network enters a solidified regime. This transition induces correlated changes across three scales: (i) a ×2–3 amplification of algebraic connectivity λ₂(G) (Cohen's d = 2.44, p = 7.6×10⁻³⁰); (ii) a shift from negative to positive Ollivier–Ricci curvature (p = 10⁻¹⁷, d = 2.0); (iii) proportional acceleration of diffusion and synchronization dynamics. The causal mechanism is direct: adding a triangle to any fragile edge increases λ₂ in 100% of cases (n = 383 trials). Validated on 47 real biological networks, the framework stratifies oncogenic signaling networks (Fragility Index = 19%) from inflammatory disease controls (FI = 88%) with Cohen's d = 10.94 — a separation invisible to standard spectral analysis (λ₂ alone: p = 0.26). The transition rests on a machine-verified bound τ(G) ≤ λ₂(G) (Lean 4 proof assistant, github.com/zeekmartin/topostability-lean4). Submitted to Journal Nature Communications.