The Topology of Verication Determines System Stability: An Epidemic Model of Recursive Test Degradation in Software Networks

Generative AI has made software artifact production structurally exceed human verification capacity, yet no existing model captures how the recursive degradation of verification itself—test code verifying production code that is itself unverified—governs system-wide stability. We model this dynamics as an extended SIS epidemic on a network, with a verification-source map δ: I I encoding verification's recursive structure, and prove that δ-topology is the decisive factor: acyclic δ-structures with irreducible dependency graphs guarantee global asymptotic stability (via cooperativity and Smith's theorem), while δ-cycles undergo transcritical bifurcation at the sharp threshold ρ_k = 1 (established for isolated cycles; the interaction with dependency propagation is characterized by a unified spectral criterion). This topological dichotomy is the paper's central result. Further results include a unified spectral stability criterion ρ(Γ⁻¹M) < 1, spectral attenuation under modular architecture, and a Wright-Fisher SDE extension yielding variance saturation with spectral divergence and three early-warning indicators. The theory provides a mathematical criterion for when AI-augmented development pipelines remain stable and when they undergo catastrophic collapse.