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. The first is that quantum mechanics is granular: every quantum state is a record of L bits, L = 6.4×10⁶¹ a universal constant, on a ring of L Planck cells, and each record's phase is hidden. 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. From the first, identical cells make momentum conserved at a vertex, and the hidden phase, respected cell by cell, makes the ring's shift carry a link phase: a gauge field, coupled to the current at a vertex with three legs. A vertex with k legs has momenta spanning k−1 dimensions; Palmer's rule that every probability is a multiple of 1/L gives such a momentum L^(k−1) labels and the sphere L^(n−1) cells; equating them gives n = k, so every elementary vertex has the same number of legs, and the three-leg one makes n = 3. Ref. [2]'s Planck pixels, defined by the horizon entropy, 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. An elementary vertex with four legs, or an energy-dependent speed of light, would falsify it.v8: The two-body premise is removed. Conservation of momentum at a vertex, the existence of a three-leg vertex, and the rule n = k (space has as many dimensions as an elementary vertex has legs) are derived from the two postulates, the last through the hidden phase of each record, which forces a link phase on the shift and hence a gauge coupling. Added: the register as a reversible automaton; the consistency reason why a vertex plane's orientation cannot be data; references to Wilson, Peierls, and 't Hooft. The title page now carries the concept DOI. v5 (9 Sep) remains the priority record.