The Division Algebra Tower as Holographic Chain: Shadow Symmetry, Cayley-Dickson Doubling, and the Origin of Spacetime Dimensions

We establish a correspondence between the holographic chain ℝ⁺ → S² → M⁴ → Gr(2,4)_ℂ and the Cayley–Dickson construction ℝ → ℂ → ℍ → 𝕆 of the four normed division algebras. Each holographic projection introduces a shadow symmetry—an anti-linear involution doubling the ambient dimension—identified with the conjugation in the corresponding Cayley–Dickson step. The Mellin transform implements ℝ → ℂ via s ↔ 1−s; celestial holography implements ℂ → ℍ via Δ ↔ 2−Δ; time reversal implements ℍ → 𝕆 via the conjugate Hodge star on Gr(2,4). We prove that the Plücker relation is equivalent to algebraic balance between self-dual and anti-self-dual components, and that the orthogonal complement acts on Plücker coordinates as the conjugate Hodge star with a pure phase. The chain terminates at dim = 8 because Gr(2,4)_ℂ is compact (χ = 6), providing a geometric interpretation of Hurwitz's theorem. The dictionary Δ = 2s cascades with critical parameter d/2: 1/2 → 1 → 2 → 4. Via SL(4,ℂ) ≅ Spin(6,ℂ) ⊂ Spin(8,ℂ), the framework connects to triality and three fermion generations. All instances of the ratio 3/8 in physics trace to dim(Im(ℍ))/dim(𝕆). Numerical verification of all proven claims is provided to machine precision. A new cohomological derivation is given: the Hasse–Weil zeta function of Gr(2,4) over ℚ decomposes as ζ_Gr(s) = ζ(s)·ζ(s−1)·ζ(s−2)²·ζ(s−3)·ζ(s−4), identifying the Riemann zeta function as the H⁰ factor; under Δ = 2s (forced by the Plücker weight k = 2), the Riemann functional equation s ↔ 1−s becomes the shadow transform Δ ↔ 2−Δ.