The Self-Reference Tower: α from the Vertex Rules of S³/2I

The Riemann zeta function at negative odd integers returns the icosahedral group orders: ζ(−1) = −1/12 (vertices), ζ(−3) = 1/120 (binary icosahedral group), ζ(−7) = 1/240 (E₈ roots). These are the rooms of a tower — the structures available to the geometry. At positive integers, the zeta function returns the couplings: the strengths of interaction within those rooms. The transition from divergence at ζ(1) to convergence at ζ(2) marks the onset of self-reference. At each level of the tower, the trivial representation — the self-referencing mode that maps everything to itself — does not contribute to the coupling. Removing it yields the augmentation ideal: dimension 11 at the vertex level (Klein's coefficient),…