Paper M: Fixed-Point Geometry of the Critical Line: The Wästlund Compactification and GUE Structure

We establish three geometric facts about the critical line sigma = 1/2 of the Riemann zeta function, each independent of the Riemann Hypothesis. First, the critical line is the unique fixed-point set of the Z/2Z symmetry s -> 1 - s-bar of the completed xi function: a one-line proof from the functional equation. Any Borel measure invariant under this symmetry has its barycentre on the critical line. Second, the Wästlund compactification maps the real line to a circle RP¹ with 0 and infinity identified at the north pole; the critical line is the equator of this circle, equidistant from both boundary poles, and the completed xi function has no zeros at either pole. Third, the Riemann zeros carry GUE spacing statistics that are self-similar at every scale; this self-similarity is the fractal structure on the equatorial circle, residing in the zero set rather than in the circle itself. We give an explicit comparison with the Levy high-dimensional concentration theorem, explaining why it provides geometric motivation but is universal to all spheres and does not select sigma = 1/2 specifically. We state precisely what each result does and does not imply; in particular, none of the three results proves the Riemann Hypothesis.