Icosahedral geometry of the Riemann sphere and the critical line

We observe that a regular icosahedron inscribed in the Riemann sphere with vertices at bothpoles places its remaining ten vertices on two circles at stereographic radii |w| = φ and|w| = 1/φ, where φ = (1 + √5)/2 is the golden ratio. The equator at |w| = 1 is their exactgeometric mean. Under the identication of the equator with the critical line Re(s) = 1/2 , thefunctional equation s ↔ 1−s exchanges the two golden shells while xing the equator. Thedual dodecahedron places four of its twenty vertices exactly on the equator. In the s-plane,the two icosahedral shells map to circles centred at s = φ and s = 1 − φ, separated by √5,with the critical line as their perpendicular bisector. The vefold rotation of the icosahedroninduces a Möbius transformation that preserves the critical line exactly. We connect thisgeometry to Klein's icosahedral equation (1884) and to Dyson's proposal (2009) that theRiemann zeta zeros form a one-dimensional quasicrystal with hidden icosahedral structure,and formulate a testable prediction regarding the distribution of zeta zeros at icosahedralangles on the equator. We then report that this direct angular test fails: the Möbius mapthat identies the equator with the critical line compresses all zeta zeros into a vanishinglysmall arc, rendering the icosahedral angles unreachable as pair dierences. The geometricobservations remain exact; the connection to the zero distribution, if it exists, must be soughtin the spacing statistics rather than in angular positions.