The Hopf Fibration of the Zeta Spacetime: Rotating Sinusoid, Topological Winding, and the Cosmic Information Density at Non-Trivial Zeros

This paper identifes the geometric structure underlying the series [1, 2, 3, 4, 5] as the Hopf fibration π : S3 → S2. A rotating sinusoid eit ∈ S1 ⊂ C sweeps the 3-sphere S3 ⊂ H (quaternions), whose projection under π is the Riemann sphere S2 = C ∪ {∞}. The critical line Re(s)= 1/2, which compacti es to the equator of S2, is the image of the Hopf bration restricted to the critical geodesic. Non-trivial zeros ρn of ζ(s) lie on this equator (under RH) and correspond to topological winding events: the count N(T) of zeros up to height T is the accumulated Hopf winding number, recovering the von Mangoldt formula as a topological identity. We further identify the Temporium of Cosmic Information Theory (CIT/TIC) [6] with the Gaussian curvature K(s) of the pseudo-Riemannian zeta metric gij: by harmonicity of Φζ = −log|ζ|, the Temporium vanishes identically except at the zeros of ζ, where it concentrates as a distribution of point charges: τζ(s) = Pn cnδ(s − ρn). The null constant t∗ ≈ 5.5612 is the phase-transition point of the Temporium: below t∗ the Temporium eld is in its free propagation phase (spacelike geodesic, no zeros, no charges); above t∗ it is in its captured phase (timelike geodesic, all zeros, all charges). We state explicitly the single remaining open problem of the series: the proof that the Hecke operator T2 restricts to □g on the critical strip, which would complete the chain of implications from the binary set {0,1} to the Riemann Hypothesis.