Could it be so easy? (Why is space three-dimensional?)

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The trouble with this sort of question is you have to be careful in your phrasing.
Maybe we observe 3+1 dimensions is because the other dimensions are curled up too tight for us to notice?
The paper depends on a model of how the Universe evolves, making assumptions about critical factors. The model may be wrong.
As the authors state right out, it is the general approach (finding dimensionality from an underlying physical model) that is different here.
No doubt others will look for other models to get other results.

I do remember seeing a chart for the kind of laws of physics you can get with n space-like coordinates and m time-like coordinates ... seem to have lost it.