Lorentzian Frame Orientation Uncertainty Below the Planck Scale: Heisenberg Uncer- tainty, Feynman Path Integrals, and Quantum Entanglement as Consequences of the Hyperbolic AdS Geometry

Paper LXIV established that the Lorentzian signature of spacetime arises from the octonionic algebra of H+: the real unit e0 gives the time direction while the imaginary units e1,2,3 give space, and Lorentz boosts are hyperbolic rotations in H3 because they mix real and imaginary directions. We now show that the uncertainty in which octonionic direction is time at sub-Planck scales the Lorentzian frame orienta- tion uncertainty δφ(E) = (ℓP /λi) ln(mPl/E) from the Fano delocalization of Paper LXII is the single microscopic origin of the three deepest structures in quantum mechanics: (1) The Heisenberg uncertainty principle. If the time direction is uncertain by rapidity δφ, the position uncertainty is δx = ℓP δφ and the momentum uncertainty is δp = mPlc/δφ, giving δx δp = ℓP mPlc = ℏexactly. (2) The Feynman path integral. A path in Minkowski spacetime is a sequence of local Lorentzian frame orientations φ(t) ∈H3. The classical path is the geodesic in H3 that extremises the AdS action. The sum over all paths is the sum over all H3 orientation sequences, weighted by the phase ei×(AdS geodesic length/ℓP ). The imaginary unit i in the Feynman weight is the imagi- nary rapidity that Wick-rotates the Lorentzian H3 to the Euclidean sphere S3: complex numbers in quantum mechanics are the real-plus-imaginary rapidity structure of H3. (3) Quantum entanglement. Two particles created at the same Fano lattice event share their Lorentzian frame orientation at creation. As they separate, each develops an independent orientation, but the correlation between their orientations persists through both the topological brane memory thread (Paper XXXIV) and the bulk 5D AdS path connecting their subsequent bulk excursions. Measuring one particle's frame orientation (position or momentum) determines the correlated orientation of the other, violating Bell's inequalities because the correlations have a non-local bulk component outside the Minkowski light cone. Brane-to-bulk excursion probabilities are the Gaussian distribu- tion over frame orientation uctuations; quantum tunneling is an imaginary-rapidity bulk excursion; the Hartman eect is the xed length of the AdS geodesic independent of barrier width. Four new predictions follow (Predictions 127130). 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.