Exact Nilpotent Collapse of Born–Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series

We identify a class of finite quantum systems — acyclic systems whose transition graph is a directed acyclic graph (DAG) — for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G₀(E)V. If T^(m+1) = 0, then the exact solution of the Lippmann–Schwinger equation is the finite sum |ψ⟩ = Σ Tᵏ|ϕ⟩ (k = 0 to m), with no condition on ‖T‖. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A₄ = t₄₂t₂₁ + t₄₃t₃₁, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born–SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric — the Born–SON depth — quantifying the intrinsic complexity of an acyclic quantum system. The algebraic core of the framework has been formally verified in Lean 4 with Mathlib (https://doi.org/10.5281/zenodo.20113599). MSC 2020: Primary 81U05. Secondary 47A10, 05C20, 81Q10, 15A16.