Basel as Universal Basin Law on the Allen Orbital Lattice

This paper derives the Basel invariant as a structural consequence of basin dynamics on the Allen Orbital Lattice (AOL) within Pattern Field Theory (PFT). Using a synchronized framework of traction, depth-weighted evolution, QUART symmetry reduction (144 to 137 admissible states), and spectral operator structure, the work establishes a discrete shell model in which: - shell populations follow the law N_n = 6n- transport scaling yields eigenvalue growth proportional to n^2- inverse response yields proportional scaling of 1 / n^2 The resulting bounded accumulation produces the invariant sum from n = 1 to infinity of (1 / n^2), equal to pi squared divided by 6. This result is derived as a consequence of discrete basin structure rather than as an isolated mathematical identity. The paper introduces no new primitives and operates entirely within the established Pattern Field Theory framework. It provides the mathematical closure linking lattice geometry, spectral scaling, and bounded global response.