The Periodicity of the Triad: Real Clifford Algebras mod 8, Vector Fields on Spheres, and the Second Face of n = 3

This paper places the base of the triad, Cl(2,0), exactly in the periodic table of real Clifford algebras, whose types recur with period eight (Bott periodicity). The periodicity is not left at the level of citation: one full cycle is constructed in explicit signed permutation matrices — from the octonion left multiplications to Cl(0,6) ≅ M8(R), tensored with Cl(2,0) = M2(R) to reach Cl(8,0) ≅ M16(R), the anchor of the period, and one step further to Cl(10,0) ≅ M32(R) = M2(R) ⊗ M16(R), an instance of Cl(n+8,0) ≅ Cl(n,0) ⊗ M16(R). At the step Cl(2,0) → Cl(3,0) the complex structure is promoted to the center: in Cl(2,0) the twist Δ satisfies Δ² = −id but is not central; in Cl(3,0) ≅ M2(C) the volume element ω = e1e2e3 satisfies ω² = −id and commutes with every generator, so the center becomes C. The same mod-8 arithmetic governs the geometry of spheres: the maximal number of linearly independent tangent vector fields on S^(N−1) is ρ(N) − 1 (Radon–Hurwitz; maximality by Adams), determined by the 2-adic valuation of N through 2^b + 8a. For odd N, ρ(N) = 1: not a single field can be combed onto S². Complete frames of three fields on S³ (quaternions) and seven on S⁷ (octonions) are constructed and verified symbolically. The "two" of the isolated antipodal fixed-point pair of the spherical-realization paper is identified with the "two" of the Euler characteristic: the Lefschetz index of each pole of the quarter turn is +1, summing to 2 = χ(S²), and for all N, ρ(N) = 1 ⟺ N odd ⟺ χ(S^(N−1)) = 2. This is a second face of n = 3, not a second derivation: the requirement of isolation remains the sole external input, and the route to the inverse-square law is only the indirect one through that paper's conditional corollary. No new dynamics is introduced. All theorem-level claims are machine-verified (46/46) by the accompanying script.