"Any math that involves a recursive algorithm dealing in hashing / integers can be done more efficiently with a GPU. While heavy memory usage can lessen the advantage, it will never be enough to exclude GPU."
Assuming you tie the staking rate to block creation, and instead of a minimum age for staking reward you implement a *maximum* age while abandoning the "weight" criteria, and offering *no* reward for successful hashing of a block, while setting a constant time for new blocks, regardless of actual completion time, you create a currency that:
Will mitigate any and all advantage that comes from successful hashing, thereby making CPU v GPU v ASIC a moot point.
Coins can "stake" almost immediately, and be set to only offer a reward in proportion to their lifetime, but as they have a maximum age, once it reaches x reward, it will no longer stake. New coins will, of course, have their own lifetime, and all transfers will effectively "reset" the coin age.
Hashrate is irrelevant, as a 1TH ASIC and a 1MH CPU both provide the same value, having 0 reward for their work, outside of maintenance to the network. If the difficulty is low enough. The only advantage speed offers is to the overall network. This could be in the form of random distribution for rewards (a lottery system, not tied to speed) or all mining power going to transaction overhead. Additionally, just do away with PoW all together.
Faucets could be used effectively, as they would be giving not only coins, but potential speed as well. If the network had 0 transaction fees, there would be no need for minimum withdrawal limits.
So, as you can see, simply by offering an inital irresponsibly high staking rate diminishing over time, disregarding network weight in staking, having an upper bound on staking age (vs lower/infinite bound) and elimination of all block rewards and transaction fee, you could create what is (effectively) a CPU only coin.
There are *lots* of economic / practical reasons attempting this will fail. I'll list the three biggest :
1) How does anybody ever get coins who doesn't have at least one coin to begin with?
2) Nobody will bother mining / hashing for 0 reward.
3) Anybody could just move "old" (non-staking) coins to another wallet. Hyper-Inflation would result, and be unstoppable. The only way to prevent this would be a centralized agency empowered to destroy coins, which would negate the entire point of the coins existance.
So, as I believe your question was on of purely theoretical value, My answer is Yes.
Yes, in theory, a CPU only coin is entirely possible. But the greed of Man will forever doom any attempt at implementation.
I will keep thinking about some innovative ideas to make this happen tho, and will keep the thread open if anyone wants to help