Topological redundancy as a structural constraint on ductile-brittle behavior in crystalline and amorphous materials

We show that the Fragility Index (FI) — the fraction of always-fragile bonds, defined as bonds participating in zero triangles in the atomic bonding network — provides a necessary topological condition for ductility and a sufficient topological condition for brittleness in crystalline and amorphous materials. Across twelve crystal structures spanning BCC, FCC, HCP, sp², sp³, and amorphous topologies, FI = 0% is necessary for ductility (KIc ≥ 10 MPa√m), while FI > 0% is sufficient for brittle fracture (KIc < 5 MPa√m). This regime separation holds across metallic, covalent, ionic-covalent, and amorphous bonding chemistries with bond energies spanning 0.45–4.9 eV. Classical descriptors quantify how a material deforms, but do not provide a purely structural criterion for whether local stress redistribution is geometrically possible. The FI addresses this prior question. Single-vacancy perturbations in BCC and FCC networks do not create new always fragile bonds, suggesting a finite vacancy-density threshold below which topological ductility is preserved — we do not derive this threshold here but the observed stability indicates it is non-zero. The edge-level triangle count distribution encodes information beyond the scalar FI: graphene exhibits a bimodal distribution (80% always-fragile bridges, 20% triangulated edges) that identifies crack pathways without mechanical simulation.