Paper 0: why three dimensions and a time

The first paper in the series asks the question you would ask a framework that plans to run simulated universes: why three dimensions of space? Most answers to that question are soft. This one began soft too, and the interesting thing about the paper is what had to be cut from it before it was honest.

The claim the paper used to make

The original draft argued that three spatial dimensions optimise novelty generation, and that this optimality selects our universe. That claim is false, and the audit found it quickly. A two-dimensional anyonic system, a planar topological quantum computer, maximises braiding novelty per unit of capacity: more distinguishable histories per resource than any three-dimensional arrangement. If novelty per compute were the selection criterion, the answer would be two dimensions, not three. The paper now states this plainly, because a framework whose central quantity is novelty flux density cannot survive being vague about which dimension maximises it.

What replaced it: admissibility, not optimality

The corrected claim is a conjunction. Three dimensions of space is the unique lowest dimension that is jointly admissible for four requirements at once, each of which is standard physics.

First, stable orbits. In d spatial dimensions the gravitational force falls as one over r to the d minus 2. In four or more dimensions it falls at least as fast as the centrifugal barrier, and every circular orbit spirals inward or escapes: Ehrenfest’s classical result. In two dimensions the potential is logarithmic and bound orbits cannot be non-intersecting ellipses. Only three dimensions give long-lived Keplerian orbits, and chemistry needs time, which orbits provide.

Second, sharp signals. In three spatial dimensions, the wave equation satisfies Huygens’ principle exactly: waves propagate on the light cone with no tail. In any other number of spatial dimensions, signals reverberate, smearing every communication. A substrate whose observers need clean information transfer needs the clean dimension.

Third, topological protection. The framework’s boundary defects need enough room to braid non-trivially, which two dimensions give, but also to be locally repairable against thermal errors, which needs the third dimension as depth.

Fourth, bounded computation. The number of boundary crossings separating regions of a causal graph scales as N to the (d minus 1) over d, and at d equals 3 this is N to the three quarters, small enough that local regions stay cheaply separable. The paper proves the conjunction: drop any one requirement and a lower dimension sneaks in; hold all four and nothing below three survives.

What this is worth

Notice what the theorem does not say. It does not say our universe’s dimensionality is explained. It says: given the engineering requirements the framework itself imposes, three dimensions of space is the cheapest substrate that meets them. That is a statement about consistency, and its value is as a filter. Every later paper that assumes 3 plus 1 dimensions is now backed by a proof that the assumption is not arbitrary within the framework’s own rules, and every alternative dimension proposal has a specific requirement list to defeat.

The downgrade from optimality to admissibility is, on reflection, the paper’s best result. It is easier to defend a theorem whose scope you have cut to exactly what you can prove, and it leaves the framework honest about the strangest consequence: the dimension that maximises novelty is not the dimension we live in, and the framework needs orbits and signals more than it needs maximal novelty per cubic Planck volume.

The next paper takes that admissibility result and asks what novelty itself is, once it is a measured quantity rather than a slogan: Paper 1, novelty as an exact resource theory.