Paper CXXXIV: The Generalised Uncertainty Principle as G2 Non-Associativity Expansion: Planck Floor, Hurwitz Ceiling, and the Uniqueness Theorem of Octonions

We show that the Generalised Uncertainty Principle (GUP) tower — the infinite series of Heisenberg corrections at the Planck scale — is identical to the Baker– Campbell–Hausdorff (BCH) expansion of the octonionic associator ei, ej, ek = (eiej)ek −ei(ejek) in the algebra G2 = Aut(O). The leading BCH term recov- ers the Heisenberg uncertainty principle exactly. Higher-order GUP corrections correspond to successive octonionic associator corrections. The BCH series diverges at the Planck scale — establishing the sub-Planck prohibition (Paper CX) as a the- orem, not an axiom. The V 2 + D2 length-dependence of Prediction P188 (Paper CXII) is the Borel resummation of this GUP tower in the G2 representation. We then prove the Uniqueness Theorem: the Planck floor h = ℓP mP c (minimum brane phase-space area) and the Hurwitz ceiling (sedenions beyond O have zero divisors) intersect exactly and uniquely at G2. The universe could not have been otherwise. 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.