Paper B: Spacetime, Gravity, and the Planck-to-Weak Hierarchy from the Split Octonion

The split octonion 𝕆_s = ℍ_s ⊕ ℍ_s·ℓ has two four-dimensional halves. This paper shows that the first half is Minkowski spacetime — its norm has signature (+,−,−,−) and its automorphism group is exactly SO(1,3) — and that both halves together contain gravity: the single product P_μ · P_ν = δ_μν·e₀ + M_μν simultaneously encodes the flat metric (symmetric part) and gravitational curvature (antisymmetric part). Gauging the resulting de Sitter algebra SO(1,4) via the MacDowell-Mansouri construction yields Einstein's equation with a positive cosmological constant; the sign of Λ is derived, not assumed. The ratio ε = 6/26 = 3/13 — the fraction of the 26-dimensional traceless Jordan arena H₃⁰ accessible to the Lorentz group — predicts sin²θ_W = 0.2308 (0.19% from the PDG 2025 value) and the Higgs VEV v = 246.21 GeV (0.006% from experiment), resolving the gauge hierarchy geometrically: v/M_Pl = ε²⁶/√2 ≈ 2 × 10⁻¹⁷, with every factor derived from the algebra and zero free parameters. The Dirac algebra Cl(1,3) is shown to emerge from the left-multiplication operators of 𝕆_s without Wick rotation, and the Dirac equation is proved to be the unique first-order Lorentz-covariant evolution they admit. The geometric seed of parity violation — the G₂ branching 7 → (2,0) + (1,1) that selects SU(2)_L — is established at the bilinear level. All results are classical; quantum behavior is Paper C. This is Paper B of the E₈₍₈₎ Split Form Unification series (Papers A–I). Paper A constructs the algebra; Paper C derives the Born rule and quantum dynamics.