Paper CXXVII · One-Octonion Brane-Bulk Framework The Theorem of Hyperbolic Necessity: The Anti-de Sitter Parent Is Forced, Not Assumed

The Five-dimensional Anti-de Sitter spacetime of the One-Octonion Brane-Bulk Framework has been, since Paper I, a mathematical assumption: the parent spacetime in which the two quaternionic black holes H and H annihilate to nucleate our four-dimensional brane. We prove here that this assumption ⁺ ⁻ is unnecessary. The 5D AdS parent is the unique geometry consistent with the empirical observation that matter exists. The proof is a chain of eight theorems connecting 'matter exists' to 'parent curvature κ = −1': (1) Matter exists (2) baryon asymmetry δM/M ≠ 0 ⇒ (3) CP violation (4) δ_CP has a definite sign ⇒ (5) Seifert surface of B(1,4) is oriented (6) σ(G) = −7, Arf(B(1,4)) = 1 ⇒ (7) Non-trivial G₂ framing of ⁷ (8) Parent geometry κ = −1 (AdS) ℝ ⇒ Each implication is either an observational fact, a theorem from earlier in the series, or a theorem from the mathematical literature (CPT, Clauwens 1987, Bott periodicity, Hill-Hopkins-Ravenel 2009). The 5D Anti-de Sitter spacetime is not selected by an anthropic principle. It is forced by topology. The Anthropic Selection Principle is replaced by the Theorem of Hyperbolic Necessity. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.