Shell Closure, Cone Geometry, and the Derivation of Cn = 2n2

We derive the shell capacity sequence Cn = 2n2 from a geometric definition of shellsvia the shell-closure condition on a spiral corridor traversing a cone interior. The closurecondition determines two geometric quantities: the step fraction δn = 1/n (the fractionof a full revolution represented by one step at the n-th shell) and the cone deficit angleαn = 1 − 1/n (related by αn = 1 − δn). These imply Nn = n tile positions per revolution.The derivation requires one additional structural assumption: that the n-th shell containsexactly n concentric revolutions. This assumption is stated explicitly and its geometricorigin is discussed; its derivation from first principles is deferred to subsequent work. Giventhis assumption and the two-state perpendicular structure of the physical class, the closurecondition yields Cn = 2n2 for n ≥ 2, matching the electron shell capacities of the chemicalelements and the prior algebraic derivation in the literature.