However, we have scanned over tens of thousands of shares worth of logs and found the real-world proof that this simply isn't the case.
Instead of
bolding an unsupported claim, how about presenting this real-world proof that sha256 has been broken, and certain hashes are acquired more rapidly than others?
There is no reason that a SINGLE USER should be generating the same amount of traffic as the rest of the pool, and no offence to you, because you've done many great things for the bitcoin community and I respect both your knowledge and opinions, but in this instance, I'm going to have to politely request that you let us run our pool the way we choose to run it.
Did I ask you to change anything? No. Simply stating the facts.
While not holding data to prove one way or the other, I must concur with jgarzik for the simple fact that unless there is something predictable about sha256, and thus we should all be working towards replacing it for bitcoin usage, the random nature of the calculation means that historical data bears no weight into future prediction.
This is just another way to say that just because you've gotten heads 10 times in a row, doesn't mean your changes of getting heads for the 11th time is bigger, assuming a well balanced coins and a good, strong throw. It is always 50%, really.
If you can in fact prove that you found a pattern on the hash calculation towards H==0, that is a potential attack vector and should be investigated. Please share the real world data you speak of!