You have to have purchased significant stake in order for a low performance hardware to be able to generate blocks.
Such a significant stake is prohibitively expensive. So both your statements are possible at the same time.
Money -> stake -> blocks vs Money -> hashrate -> blocks
PoS: The mathematical properties of staking is distributed according to stake using statisical means
PoW: The mathematical properties of mining is distributed according to hashrate using statisical means
both stake and hashrate are purchased with the same money. If to use your level of math proofs, I have proved that PoW and PoW are identical! But I am not saying that. each has its own place where it has advantages over the other
But please, let us not just be making unsubstantiated statements, this would become some sort of illogical "debate" where logic and truth are not part of the process. You do want to have truth and logic part of the discussion, dont you?
James
Purchasing stake may well be expensive.... Let's say, for example that purchasing enough stake to attack a POS chain costs you as much as buying enough mining equipment to attack a POW one?
So, once you are ready to attack both chains, then what?
*) In the POS chain you can attack continuously at zero cost, since block production costs you nothing
*) In the POW chain you must spend continuously to produce a valid block, and what's more, you must outpace the entire network in order to produce a longer chain
Simplifying, in a POS chain the attack cost is a constant (the price of the stake) whereas in a POW chain, the attack cost is superlinear in the number of blocks you want to produce.
It all comes down to the same thing: money
And if we were to hypothetical bitcoin vs bitcoinPoS, then I would think that it would cost a lot less to buy 50% of hashrate than 50% of marketcap.
You need to take into account that prices are not static and any pre-attack accumulation will make the price go up exponentially.
And it seems that you are admitting that in the absence of an attack, there is not much difference. So the difference is the cost to conduct an attack.
So, what would it cost to buy half the hashrate for bitcoin?
James
P.S. Another twist that makes this not so trivial is that once the attacker has accumulated a large stake, he has a large stake! And now his decisions to attack need to take into account the cost to his stake.