Write a simulation where you have n entities with an equal amount of resources. Now choose two entities at random and let them "trade", where the outcome is random but fair. You will get a "wealth distribution" that is not a gaussian normal around the average but more like a poisson distribution. And this is even with no difference in competence of the market participants.

You can add various corrective terms to make get "equality of outcome", but to get to a gaussian distribution around the average you need really drastic/totalitarian measures.

The best system from a pure math POV would be a flat tax combined with universal basic income. But unfortunately it is naive to assume that this is politically viable.

Politics largely is selling a particular group an advantage over the general population in exchange for votes. And a flat tax / basic income system is just too simple for that.

The distribution looks like it does because you often can't substitute quality with quantity, so the money gets dis-proportionally funneled to the top few. So for example if we in the future figure out good ways to collaborate so ten average people can replace a top CEO then you'd see CEO pay go down the drain.

Depends on the kind of system obviously.

Power law is reasonably easy to get. Even with identical starting wealth just having chance + compound interest is enough to produce it. No need for anything more complex than that.

Well for starters, such a curve would be fundamentally impossible. You draw such a curve by sorting the data points from wealthiest to least.

But I suppose, adding no additional logic you could have one half of a normal distribution, normalized by multiplying everything by 2 to account for that.

However, that doesn't strike me as a very natural distribution for a self-organized system with some randomness, and exponential growth. Power and wealth always seems to find its way to some type of power law distribution. You could only enforce such a distribution by concentrating power to the most extreme limit of a power distribution to exert control over the rest. And it's likely that for any definition of wealth you could come up with, people would just find a proxy for wealth that isn't counted by which they could start to exert more control.