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: it follows from two postulates and some counting. The first is that quantum mechanics is granular: every state is a record of L bits on a ring of L Planck cells, L = 6.4×10⁶¹ universal, each record's phase hidden. The second is that the momenta an elementary interaction can carry have one address space, the directions of n-dimensional space, n unknown. All else is derived: interactions, their legs, one momentum per address, that the address space is the cosmological horizon, and n. From the first follow momentum conservation, a gauge field from the hidden phase, and an elementary event with exactly three legs, one mark crossing one link with one quantum; a four-leg event is two crossings, coincident at most once in L. Three momenta summing to zero are coplanar for n ≥ 2, so an event needs a direction on one ring and a magnitude, L² labels with no n; Palmer's rule, every probability a multiple of 1/L, gives the directions L^(n−1) addresses; finite L and the gauge clause fill them one momentum each, so n = 3. Planck pixels from the horizon entropy (Ref. [2]) agree to a factor π; in real n the root is 3.008; 2 or 4 would need a count wrong by at least 3×10³⁰. The horizon is one, so space is one. A four-leg event that is not two crossings, or an energy-dependent light speed, would falsify it. v11 (12 Sep): Postulate 2 is restated as an address space (the momenta an elementary interaction can carry have one address space, the same for every interaction, whose addresses are the directions of n-dimensional space), and what it formerly asserted is derived: the sphere, from the momentum doublet being a qubit and from no distinguished cell or direction; every great circle a ring; and one momentum per address, at most one from finite L (n ≥ 3), none empty from the gauge clause and the uniqueness of the address space (n ≤ 3). The only word still postulated is that the address space is one. The three-leg vertex is exact: a crossing shifts a link's flux by one unit (Kogut–Susskind), so a four-leg event is two crossings, coincident at one cell at most once in L. The L²/2 accounting is withdrawn, since the sign of the polar digit is the sign of the energy; the root is 3.008 uniquely and π is the whole residual. Field theory's four-point contact terms are recorded as open. Corrected: the sign of σ_y, "L² cells of area h," the orientation lemma's n ≥ 1, Sec. IV's opening inference, and stale citations into the register. Removed as unused: reversibility and the umklapp energy claim. v5 (9 Sep) remains the priority record.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.