Phase-4 Bridge Closure: Molecular Binding and Copper Nonlocal Response from Protected Geometry

This archive contains the Phase-4 bridge-closure manuscript and its reproducibility materials. The paper extends the local Phase-3 closure theory to the first protected nonlocal bridge sector. Phase 3 closes chromium as a strict local term object, while copper remains with a positive one-center residual. This residual is interpreted as the first Phase-4 bridge threshold, not as a failure of the local theory. The Phase-4 scalar response sector is then applied to two minimal closure tests. First, the hydrogen molecule is closed by the protected scalar stiffness bound, with the numerical certificate LambdacritH2 = 3. 103433543709 and Lambda₁ <= 1. Second, copper is closed as the first canonical Phase-4 bridge object by the minimal Riesz response associated with the Phase-3 copper threshold. The accompanying notebook reproduces the numerical certificate and generates the output archive. The generated files include the Phase-3 boundary certificate, the H2 scalar closure scan, the canonical copper bridge certificate, the scalar Cu certificate, a radial representative of the canonical source, and the final Phase-4 closure-status table. The main closure result is: Cr local closedH2 molecular closedCu bridge closed Thus the archive documents the minimal geometric and numerical closure of Phase 4.