Verificationthat a hash output is valid for a given inputshould be orders-of-magnitude more efficient than computing the hash.
The validation ratio when used in a proof-of-work system also depends on how fast the hash is, because validation only requires one hash whereas solving proof-of-work requires as many hashes as the collective difficulty or more importantly the pool's minimum share difficulty.
Cuckoo Cycle is not a hash.
The Cuckoo Cycle hash [19] significantly increases the asymmetric validation ratio, but has unprovable security because its algorithm is based on permutated orderings which do not incorporate diffusion and confusion [20]
Diffusion and confusion are properties of hash functions. Cuckoo Cycle is not a hash function.
You need to get your basics straight.
Although its entropy is large, i.e. the factorial permutation of all buckets in the hash space, it doesn't require that its entire memory space be accessed, thus possibly bit flags could be employed to reduce the memory used and make it faster.
There is no "entire memory space". Cuckoo Cycle uses memory for data structures, not for filling up with random garbage like scrypt does.
I agree that Cuckoo Cycle can be parallellized. In fact the GPU solver works best with 16384 threads.
But it's only marginally faster than 4096 threads. Because they're saturating the random access
memory bandwidth.