The Dimension of Space from the Granularity of Hilbert Space

We derive the three dimensions of space from two postulates. The first is the register: every qubit is a string of L bits around a ring of L cells of one Planck length, with L = 6.4×10^61 universal. Its bits are the outcomes of one measurement and its position is the phase, and each cell counts phase from its own zero mark, on which no law depends. The second is that there is a sphere in d-dimensional space, d unknown, every great circle of which is such a ring, and its cells label the momenta an elementary interaction can carry, one cell per momentum. An elementary interaction is an event at which its legs exchange momentum. Momentum is conserved, so the momenta of an interaction with k legs span k−1 dimensions, and each of them is one of L^(k−1). The sphere has L^(d−1) cells, so d = k for each interaction, and d need not be universal. In the register a moving string can pass one step of phase to the offset of two neighboring cells. That event has three legs, so d = 3. The offsets are light. An observation is an exchange of momentum, and a string's momentum changes in no other way, except when two strings meet at one cell, which for uncorrelated strings happens once in L pairs. Every observation is therefore made with this event. A four-leg interaction, rare or common, would be seen only through it, and its four momenta span three dimensions, so it would appear three-dimensional. With L from the cosmological constant the sphere is the horizon of an observer at the interaction. A count relaxed to real d finds no other solution. v16: Two-body premise dropped: with a zero mark of phase at every cell, conservation and the register's three-leg interaction are derived; the third leg is the offsets between cells, identified with light. Dimension counted per interaction: k legs need k dimensions, so d = 3 for the register's interaction, and every observation is made with it. Independent of the gravitation and vacuum papers; L fixed from the observed Λ, and the count does not use its value. Prior qubit-based derivations of d = 3 engaged in the introduction. Capacity 2^N ≤ L derived here; n renamed d; closing ledger of what is postulated, derived, and open. v5 (9 Sep) remains the priority record.