An interesting and difficult question alexbalex. As with a lot of these things one could say yes as for as normal life is concerned but then say no when we stray into the quantum domain and finally maybe or maybe not! when we think a bit more about it.

Firstly let me say a bit about scale invariance. Put simply this means that on whatever scale you look at something the larger characteristics of the thing are on the whole the same. However different disciplines tend to stress what they mean by characteristics. The classic physical or mathematical case is a fractal structure where a graph or a coastline has broadly a similar degree of wiggles on a wide range of scales. In a mathematical equation (for example the Mandelbrot set) this can go on to infinity but in physical structure this can break down as the properties of a material change significantly with the scale on which you observe it. In the case of physical laws this means that the laws apply on whatever scale you use to observe them.

Scale invariance also has a meaning in quantum theory in that the strength of the interactions between particles is the same on whatever scale these things are observed.

However when we come to think about quantum theory properly, things as always can get a bit fuzzy because quantum uncertainty come into play. This sets clear limits on what we can observe precisely on individual experiments. We can only know what happens on the average from a large set of experiments so at high energies, and by implication small scales details, of lower energy interactions are in effect hidden. This of course means that scale invariance may be true, but then again may not as long as the statistics is right for us "outside the box" defined by the uncertainty limits.

Now the really important result of this is that gravitational interactions between particles are hidden way down inside this box and we could only observe gravitational interactions at vast energies way beyond any sort of energy even vaguely conceivable in a scientific experiment. However in the very extreme conditions when a the core of a black hole is contracting inside its event horizon towards this theoretical mathematical "singularity" everything is squashed together so tightly that normal charges between positive and negatively charged particles are effectively neutralised by the extreme proximity of particles with opposing charges and the strong forces do not come into play leaving only gravity and possibly the ideas generated by the vast array of string theories to operate. This is where the fundamental properties that define a universe are probably defined. This may be fixed by scale invariance, things settling out "randomly" or by some sort of evolutionary or "crystallisation" process. This is where the quantum gravity and cosmological theoreticians meet.

Unfortunately the cosmologists do have their observational limits currently mostly associated with the cosmological microwave background which is light emitted when our whole universe was like a single exploding star and the recently published observation of polarisation due to gravitational waves on this background and possibly a future experiment that may be able to observe a cosmological neutrino background radiation.

This may be an insoluble problem but I reckon that there is a way to look at this problem but to talk more about that that would go beyond the limits of this page which is limited to currently widely accepted theories.