Proof of stake seeks security without cost,
but the laws of economics say that security
costs will always rise to meet the rewards...
Laws of Economics are not universal, they are environment dependent. Change environment (this is what cryptos do) and you will change the laws.
Right now we observe that there are some PoS systems that consume almost no electricity but have not been successfully attacked yet (and the reward is very appealing). Your statement contradicts to observations and this makes me suspect that you are wrong...
...And if you don't want rewards to secure the
system, then you'd be taking away one of
the most important aspects of Bitcoin's security model,
which is incentivizing people to honestly participate
in the network instead of attacking it.
Noone proved that Bitcoin's security model is the only viable one.