The Dimension of Space from the Granularity of Hilbert Space

Why is space three-dimensional? We now know: because of the two postulates of quantum granularity. The first: every qubit is a string of L bits on a ring of L Planck cells, L = 6.4×10⁶¹ universal, with a phase on every link. The second: a direction is the state of one qubit. A qubit's state is a bounded count and a cyclic turn, which close to a sphere of L² cells when no cell is distinguished. The directions of n-dimensional space have Lⁿ⁻¹ cells, one rational probability per angle. The two sets are one, so n = 3. A momentum is a direction on one ring and a magnitude, L values each, so a vertex's momenta have L² labels with no n among them. Finite L and the gauge clause fill the sphere with those labels one to a cell, so every elementary vertex has three legs, a prediction the register's own interaction meets. Momentum is conserved, a four-leg event is two three-leg events, everything that interacts with light shares one sphere, and it is the cosmological horizon. Could n be a non-integer? The grid exists only for integer n, but the horizon's Planck pixels count for real n, with one root, three within the pixel error, 0.008 at the physical L and zero as L → ∞; 2 or 4 would need a count wrong by 3×10³⁰. A four-leg elementary vertex whose coupling is not the square of a cubic one, or an energy-dependent light speed, would falsify it. v14: Postulate 2 is now "a direction is the state of one qubit," and n = 3 is counted from a qubit's two digits; every elementary vertex having three legs is a prediction, with four-leg events as two of them, and the register's own interaction is the check. Postulate 1 carries the phase on every link and the hidden phase of a state. The Planck pixel enters only to continue the count to real n, where its one root tends to 3. Consequences for the register updated to its current version; language revised throughout. v5 (9 Sep) remains the priority record.v12: Postulate 2 weakened by one word: its addresses are directions of n-dimensional space, not the directions. That every direction is an address is now derived, from the gauge clause and the one address space. Two words of Postulate 2 remain postulated, one and directions, correcting v11's "only one." Citation removed from the abstract. Wolfram's hypergraph-rewriting models added to the introduction as a third prior approach, one that leaves dimension open. v11: Postulate 2 restated as an address space; the sphere, every great circle a ring, and one momentum per address (n ≥ 3 from finite L, n ≤ 3 from the gauge clause) are derived. The three-leg vertex is exact: a crossing shifts a link's flux by one unit, so a four-leg event is two crossings. The L²/2 accounting is withdrawn; the root is 3.008 and π is the whole residual. Minor corrections and unused material removed.v8: The two-body premise is removed; conservation, the three-leg vertex, and n = k are derived from the postulates through the hidden phase, which forces a gauge coupling. References to Wilson, Peierls, and 't Hooft added; concept DOI on the title page.