Ehrenfest and Weyl showed this as early as 1920 and 1922, respectively.
3+1 is also baked into the Minkowski metric by the latter's definition, which leads to the Poincaré group being the isometry group on local patches of spacetime; the Poincaré group is a subgroup of the Standard Model, and is extremely well tested in controlled, laboratory settings (and supported by an abundance of observational evidence too).
Extra dimensions of spacetime have to be small compared to the Planck length in order not to be obvious today. For all practical purposes, if such tiny dimensions exist, we can safely omit them from effective theoretical descriptions of everyday physics, just like we can ignore things like very large extra dimensions where the smallest step one can take is much bigger than the Hubble diameter.
http://nautil.us/blog/the-universe-expanding-symmetrically-a...
Small regions of our spacetime are locally Lorentz-invariant, meaning that a number of observable quantities (magnitude of angular momentum and mass are two) do not change for a particle held at the origin of a system of coordinates as the particle is arbitrarily rotated or boosted.
Arbitrariness is important. If we are observing a distant and predictable multifrequency radiator and define spherical coordinates with the radiator at the origin, then any movement our observation gear makes that isn't exclusively radial is equivalent to a rotation of the distant radiator. You can do this by holding the radiator at the (spacelike) origin, and the observer at its fixed (spatial) coordinates; in order to keep these coordinates constant, you have to rotate the system of coordinates to counter wholly non-radial relative movement. At large distances, the rotation at the origin becomes extremely small. Small or large, Lorentz invariance means the radiator has the same mass (it's by definition rest mass since it's always at the coordinates [0,0,0,t]) and momentum.
So arbitrarily small rotation goes hand-in-hand with arbitrarily distant observers, and also with nearer observers who can displace themselves tiny amounts.
So is there a minimum rotation?
The arbitrariness of rotation is a in conflict with "cell shape", as when your "cell" distinguishes between edges and faces, the discrete nature of rotated mass-energy-momentum exposed through the (corner-filled, discrete) cell structure leads to different observables under rotation compared to that of the (smooth, continuous) spacetime of (either theory of) relativity. This deviation is larger for objects of higher momentum; and remembering Einstein's relation for rest-massless particles, E = pc = \hbar\omega = h / \lambda, that means that for different-wavelength photons emitted from the same source, the higher-frequency photons will arrive later than the lower-frequency ones. We have good observational evidence against frequency-dependent arrival times from bright distant objects (supernovas, gamma-ray bursts, even millisecond pulsars).
We can conceive of a "cell shape" which is uniform under infinitesimal rotations, but at that point you have shifted one set of infinitesimals to another, and in the context of this topic (a simulation that among other things saves on state by abolishing real numbers in dimensions of length, rotation, boost and/or translation) is pretty much a non-winner.
(Additionaly, ignoring the simulation context, you would also almost always run into difficulties if your "cell size" -- a minimum length in space or a minimal interval in spacetime -- is large enough to produce observables.)