The Spherical Realization of the Triad: Dimensional Selection by the Isolated Antipodal Fixed-Point Pair, and the Two 4π's

This paper constructs, as theorems, the triadic configuration on the sphere, inside the orthogonal group O (3) obtained by adding a single axis to the two-dimensional algebra Cl (2, 0) of the triad. The three-dimensionalized twist has exactly one antipodal pair of fixed points on the sphere and acts on the orthogonal complement of its axis as the two-dimensional twist itself (±i, no real eigendirection) ; the orthogonal reflection fixes an invariant great circle pointwise (+1, a continuum) and exchanges the fixed-point pair (−1, a discrete label). The casting is fixed by the dimension count of eigenspaces, and carries an exact duality: no exchange of roles exists inside the centralizer of the axis (z² + 1 has no real root), while the exchange holds exactly as an O (3) conjugation. The antipodal map decomposes as −I₃ = D₃²·L⊥, joining the series back to the non-orientable-spacetime paper. The fixed-point pair being isolated (S⁰) is equivalent to spatial dimension n = 3, where the total solid angle 4π steradians meshes with the spinor closure 4π radians through the holonomy exp (−iΩ/2): hemisphere 2π ↦ −1 (the residual unit), full sphere 4π ↦ +1 (the closure), the factor 1/2 being that of the double cover. As a conditional corollary, flux conservation yields F ∝ r^ (1−n), inverse-square only at n = 3. A demarcation is placed first: this paper does not derive the three-dimensionality of space. The isolation requirement is an external input; the algebra alone does not select a dimension (over real dimensions, Ω = 4π even has a second solution n ≈ 12. 81). No new dynamics is introduced. All theorem-level claims are machine-verified by the accompanying scripts (63 checks, all passing), included as a ZIP archive.