I claim this is self-evident. What counterproof do you propose?
To be fair, it's not completely self evident in the sense that monetary inequality is the only incentive to improve. There are other kinds of inequality. Social status, for instance, rewards improvement, although it's very different.
If it were self-evident, it would not need to be stated.
Imagine the game monopoly. Then, imagine a house rule that says all players receive $300 at the start of each turn.
That's not the only source of income in Monopoly though.
Yes, but the incentives improved equally.
Are you unironically claiming that having an inequality of economic outcomes is the only way to motivate/incentivise human behaviour?
No, "inequality of outcomes" . If you are motivated by status, then inequality of status outcomes. If you are motivated by helping your siblings, then inequality of your siblings' well being. If you are motivated by glitter, then just how much glitter you get. Money is but one dimension... I am saying it is self-evident that there must be some non-trivial gradient in at least one direction.