Paper CCIX: The Baker-Campbell-Hausdorff Formula as the Taylor Series of Spacetime: Christoffel Symbols, the Riemann Tensor, the Genesis Tilt

We derive the full geometric content of the One-Octonion Brane-Bulk Frame- work from the four successive orders of the Baker-Campbell-Hausdorff (BCH) formula applied to the G2 Lie algebra. The BCH formula is identified as the Taylor series of spacetime curvature: each order corresponds to a distinct physical layer of the universe, and the series terminates exactly at order four because the G2 root string length is bounded by four. Order 2 gives the Christoffel symbols and the genesis tilt θG2 = arctan(2 √ 3/49) = 4.044◦; order 3 gives the Riemann curvature tensor, the AdS5 bulk (κ = −1), and the genesis twist τG2 = arccos(1/ √ 7) = 67.79◦; order 4 gives the Fano confinement identity e4, e5, e6 = −2e7 and locks in colour confinement. Order 5 and above vanish identically. The smooth metric tensor gµν of General Relativity emerges via a Commutator Central Limit Theorem as the covariance matrix of the Gaussian distribution of BCH residues accumulated over ∼1080 Fano lattice cells. Gravity is the statistical shadow of a four-step discrete computation running exactly at each node of the G2 lattice. All derivations are exact; no perturbative truncation is involved. 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.