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RE: What would a legitimate basic income buy?
Life is the opposite of entropy. The laws of physics are not actually "laws", but simply best known description for how the world works.
I don't know what "problem" you think cannot be solved is. My stated problem is, "how to divide the world's resources fairly". That is a problem that can be solved for certain definition of "fair" being all men created equal receiving equal opportunity and equal inheritance of the natural resources of the universe.
You have identified a separate problem which is that even if everything was "mathematically fair", people may not perceive things to be fair. That is a truly intractable issue.
Entropy is a matter of perspective. Life for us living things, is not entropic but in the grander scale of things, considering the universe is a few billion years old, it is very much entropic. One can see the dessolate star systems and call them "ordered" and characterise us as "chaotic" since we are the exception to the rule.
Check human history. Any age. The gap in wealth widens as specialisation and industries grow. The reason for this correlation is because the more industries we create the more menial positions we need. The more menial positions we need to cover the less CEO's we need. This is how 1% ends up owning almost everything on the planet. Heck even in feudalism the percentage was more around 10%. In Ancient Egypt the gap between the slaves and the citizens was much smaller. Same applies with ancient Greece.
What is fair for me might not be for you. This is again because of entropy. People grow in different environments, different cultures, have different histories, customs. We cannot possibly come together by means of agreeing with something as generic as "fair".
Nonetheless, we live better lives today compared to any time in human history. The gap actually helps people since we all grow richer and even taste the upper strata once in a while.
Equally as important (and dangerous) is a) who is the arbiter of what is fair and b) who decides who is the arbiter of this fairness?
Basic survival requires food, water and shelter. Beyond that, it is necessarily subjective because there will always be perspective influencing what someone deserves and what others think someone deserves.
Apart from protecting people from starvation, thirst, disease and freezing/burning to death, I'm not sure how a system could be implemented without an opt-in charity component.