Uh, no. We are not. Both nature and nurture deal unfair cards. And markets and capitism most certainly alone do not promote solidarity. That is, in any real world scenario plaid out so far.
On the contrary, completely free markets often lead to oligopolies, monopolies and cartells.
Or is it supposed to be a bad joke in empirical economics, to start with a population with equal chances and see it tumbling into a winner takes all pyramid of distributions of wealth?
In a completely fair game we could choose not to do business with known unfair players. Real world does not work this way. As soon as there is capital surpluss, someone will get an upper hand in negotiations. The wealth allows the other party more negotiation tactics. Hence, anyone who can use the full leverage of their wealth will come on top. In the average case.
Money isn't the only route to power, of course. Politics and violence are an inseparable facet of the negotiation dynamics of our species, even if they don't affect every business deal. For example, by imposing dominion over supply trade unions can increase leverage of employees over employers, who then may under some situations use government sanctioned violence to increase their leverage - or vice versa. Etc.
I don't have a formal proof of this. It just seems to be the way the game is played.