Lorentzian Signature and the Spacetime Interval as Octonionic Algebra: Hyperbolic Planck-Scale Geometry as the Dening Mechanism for Special Relativity Paper LXIV of the Brane-Bulk Octonionic Series

Special relativity rests on two axioms: the constancy of the speed of light and the equivalence of inertial frames. It derives the Lorentzian spacetime interval s2 = c2t2 −x2 −y2 −z2 but does not explain why spacetime has signature (−, +, +, +) rather than Euclidean (+, +, +, +). We show that the brane-bulk octonionic framework provides a complete derivation of Lorentzian signature from rst principles, without ad- ditional axioms, through three mutually reinforcing mechanisms. Mechanism 1 (Al- gebraic): The Minkowski spacetime interval is identical to the octonionic norm of the H+ quaternionic sector e0, e1, e2, e3. Since e2 0 = +1 (real unit) and e2 1, 2, 3 = −1 (imag- inary units), the norm is ∥X∥2 = c2t2 −x2 −y2 −z2 exactly. The Lorentzian signature is the algebraic statement that H+ contains exactly one real and three imaginary octo- nionic units. Mechanism 2 (Geometric): Lorentz boosts mix the real direction e0 with the imaginary directions e1, 2, 3; since these have opposite squares (+1 and −1), the rotation between them is necessarily hyperbolic, generating the group SO (1, 3) with ra- pidity parameter equal to a hyperbolic angle. The Lorentz group is therefore the group of automorphisms of H+ that preserve e0, derived rather than postulated. Mechanism 3 (Holographic): AdS5 is itself the hyperboloid X2 0 + X2 5 −X2 1 −X2 2 −X2 3 −X2 4 = L2 AdS with isometry group SO (2, 4) the conformal group of 4D Minkowski space. The Lorentzian boundary metric of AdS5 is not imposed; it is the unique boundary metric compatible with the bulk hyperbolic geometry. The negative curvature R ≈−20/ℓ2 P at the Planck scale (Paper LXII) projects onto the brane boundary as the (−, +, +, +) signature of Minkowski spacetime. The light cone is the zero-norm shell ∥X∥2 = 0 of the H+ algebra a topologically protected structure whose interior (timelike, ∥X∥2 > 0) and exterior (spacelike, ∥X∥2 < 0) are algebraically and topologically distinct. Causality is therefore not a separate postulate but a consequence of the topology of the octonionic algebra. The rapidity φ of a Lorentz boost is the geodesic distance in hyperbolic 3-space (H3), and the maximum physically accessible rapidity before hitting the Planck bound- ary is φPl = ln (2mPl/me) ≈52, corresponding to the highest conceivable Lorentz factor γmax ≈mPl/me ≈2. 4 × 1022. Four new predictions follow (Predictions 123126). 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.