How does the pool know the share hashes are calculated and not cheated? Miners calculate share hashes and submit them to the pool for their proof of work. And if the share hashes leads to a correct block hash in terms of difficulty, everybody in the pool gets paid according to their submitted work using different distribution methods.

Question:
Why can't a miner just submit random numbers as fake share hashes (little work) to the pool, instead of doing the actual hard work of hashing?

So for the pool: Is there a way to prevent this without recalculating the hash to find out its fake? Recalculation is hard work, and annihilates the sense of delegating it to miners.

And if there is no control it would only be visible with statistics as the pool is too unlucky on the long term. Ex.: a miner submitting 100THash/s and never finding a match might doesn't even calculated the hashes but just submits random stuff.

Edit: Clarified question to distinguish from similar sounding questions.
If the hashes are not recalculated/checked by the pool independently 


The hashes would be checked by the pool, so you can't just submit random numbers. Pools would most likely ban miners which did that, for wasting their time.