Constraints on the Universe as a Numerical Simulation (2012)

EDIT: Thanks for the expansion! Fascinating and really interesting idea. The idea, "There is an existential difference between a mathematical truth and a specific occurrence of the 'pattern' described by that truth within the physical material of the universe" is a fascinating one.

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.

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.)

The Standard Model has as a subgroup the Poincaré group, which is the isometry group of flat spacetime, which is both the (global) spacetime of Special Relativity and is the local spacetime of General Relativity for a pretty narrow definition of local, although we regularly construct Local Inertial Frames which are regions of spacetime that are so approximately flat that the difference is negligible.

The Poincaré group has as a subgroup the Lorentz group, and its generators include rotations about the three spacelike axes of Minkowski (i.e., flat) spacetime.

The other symmetries of the Standard Model are invariant under these rotations.

A distant observer moving transversely relative to a particle observes a tiny rotation of the particle. The particle's fundamental properties do not change under that rotation, which can be arbitarily small.

Rather than descending into group theory to reason about a minimum translation or boost, we can look to the energy-mass-momentum equivalence E_r^2 = (m_0 c^2)^2 + (pc)^2 and focus in on photons so we can ignore the m_0 term. Here we have E = E_r = (pc) = \hbar\omega = hc/\lambda = hf. Photons come at arbitrary frequencies, and we have good blackbody radiators all across our sky. If there were a minimal length scale, we would expect that derivatives of position would incorporate that minimal length, and so we would impose observables on things like the doppler shift, for example if we accelerate in a straight line towards a radiator emitting extremely high energy photons.

So it's not so much "why wouldn't it have" but rather, "Q: does observation or experiment support a minimum scale that is large compared to the Planck scale for spacetime intervals in our universe? A: no"

The answers for minimum length et al. scales that are small compared to the Planck scale are subtler (essentially by definition of the Planck scale, at those scales quantum corrections to account for gravitation become significant, spoiling the observability of short spacetime intervals) but so far still 'no'. For every particle in the universe there can be an (ultrarelativistic) observer who sees the particle wavelength shrink below the Planck length; there is nothing special about this observer -- there is likewise a possible observer who sees the particle at some much longer wavelength -- and the point of relativity is that neither observer is more correct than the other. Furthermore, if the former type of observer manages to see a star ultra-blueshifted and extremely Lorentz-FitzGerald-contracted, that observer does not create a black hole; no event horizon forms (event horizons are a global feature of the causal structure of spacetime[0]), and so we can turn an argument about Planck length into an argument about localized Planck energy: there is nothing obviously special in spacetime about the Planck scale. However, our ultrarelativistic observer will certainly see some very strange stuff courtesy of the Unruh effect: different observers observe different particle counts, and one would expect our observer would see an explosion in the number of particles compared to a more typical observer, and those particles -- like all others -- will also interact gravitationally, and we don't know yet how to decomplicate the picture enough to do useful calculations on them.

[0] different observers may disagree on precisely where the event horizon is, what its shape is, and even the count and energy of the particles just outside the event horizon. but they will all agree that there is an event horizon.

I find it amusing that you think a topic as widely studied and disputed as consciousness is "open and shut" because of some assumptions that you choose to make. Nonetheless, I do appreciate hearing your perspective so thank you.