We're talking about political philosophy, an inherently subjective topic, and you're asking for "proof." That alone is evidence that you're missing the whole point. Nobody has definitive proof that any governmental system is the best--otherwise there wouldn't be any debate on the topic. It'd be settled and anyone using a different system would be objectively wrong, and yet that's not the case.
I can explain my position as to why Libertarianism is rife with problems that you don't recognize, but would it really make a difference? Since you refuse to accept any philosophical ideas without "proof" (which, again, is totally irrelevant in this context), I'm not sure how I'm supposed to explain it to you.