One clarifying point:

Even when nearly everyone gets wealthier, if inequality grows, most people don't feel like it's fair.

Here's an idea: a UBI that's pegged to GDP and population.

Make it comically small to start, like say 0.25% of GDP split evenly between all citizens. Based on 2021 US stats, that's $176 per person per year. That seems relatively unobjectionable regardless of your politics, even assuming that the % is likely to drift higher over time.

With increasingly advanced AI and cheap energy, GDP would rise over the long run independently of population, ultimately increasing the UBI to a livable salary.

Well, then we have been living in a system that's as close to fair as possible since global individual wealth has been growing for a long time. The few times we had it shrink significantly over some period of time is when local populations attempted forced wealth distribution.

Our present system, by this definition, is amazing!

Not that person, but I don’t believe that sentence approaches a definition of “fairly”. “Make everyone richer” is all fine, until you realize that you’re only a penny richer. But maybe it’s justified. That justification is what the “fairly” definition would require, and is extremely important.

Why would anyone contribute more than is required to get what they need, if they’ll never see any benefit beyond that?

I prefer a system where people that work harder get more so they can have what they want, rather than just the bare minimum to survive (which is presumably what is meant by “what they need”).

Defined by man that have never worked honest day in his life and lived off family, repeated by people that don't know human nature well enough to know it doesn't work

to each according to their need, from each according to their ability

^ communisme