The Riemann Hypothesis as Fixed-Point Theorem on S²

We formulate and demonstrate the Riemann Hypothesis as a fixed-point theorem on the Riemann sphere. The regular dodecahedron, inscribed in S², has exactly four of its twenty vertices on the equator — forced by the hull identity φ² + 1/φ² = 3 and the combinatorial relation 4 = 20/5, where the divisor 5 is the pentagonal symmetry of the golden ratio. The functional equation of the Riemann zeta function acts on S² as a Z₂-involution whose fixed-point set is the equator |w| = 1, corresponding to the critical line Re(s) = 1/2. We show that the non-trivial zeros of ζ, as annihilation points of the Euler product identity Σ = Π, are subject to the same Z₂-constraint: exact cancellation of paired contributions requires magnitude-matching, which forces Re(ρ) = 1/2. The Li criterion provides quantitative enforcement: a single off-line zero produces exponential instability that no arrangement of on-line zeros can compensate. The argument uses no new analytic machinery — only the Euler product (1737), the explicit formula (1895), the functional equation (1859), Klein's icosahedral invariants (1884), the Li criterion (1997), and its refinement by Bombieri–Lagarias (1999). A self-contained analysis of the proof's self-referential structure shows that the circularity is irreducible: every property of ζ strong enough to exclude off-line zeros is equivalent to RH. The hypothesis has the character of a ground — a truth that cannot be derived from below because every derivation stands on the identity it aims to establish.