The Dimension of Space from the Granularity of Hilbert Space

Why do we live in a three-dimensional world? It has been asked since antiquity. We now know the answer; it follows from two postulates and one premise. The first postulate is that quantum mechanics is granular: every quantum state is a record of L bits, L = 6.4×10⁶¹ a universal constant, and Ref. [1] placed that record on a ring of L Planck cells. The second is that the momenta an elementary interaction can carry are labeled by the cells of a sphere in n-dimensional space built on such rings, with n left open, integer or not. The premise is that elementary interactions are two-body, since three particles meeting in one cell in one tick is a double coincidence. A two-body event's momenta lie in a plane, and a plane meets the sphere in one great circle, so the event needs a direction and a momentum, L values of each: L² for any n. Palmer's rule that every probability is a multiple of 1/L, applied to each angle, gives the sphere L^(n−1) cells, and L^(n−1) = L² requires n = 3. Ref. [2]'s Planck pixels, defined by the horizon entropy, not the register, agree to a factor π; as an equation in real n their single root is 3.008, and 2 or 4 would need either count wrong by a factor 3×10³⁰. The sphere is one, so the space is one, and every interaction's plane lies in it. A three-body elementary collision, or an energy-dependent speed of light, would falsify it.