SatoshiDICE doesn't know the blockchain transaction ID in advance so SatoshiDICE cannot influence the results of my wager.

So it is provable mathematically that the house cannot cheat the player.

You are describing the situation where if I know the secret before I place the wager, I can cheat the house.  Now that the house has investors, you are suggesting that this list of secrets makes it is possible for a party [edit: a party who has access to the secrets] to cheat the house, lowering its profits, which in turn, lowers the dividends for investors.

Over enough trials, the chances of that occurring without notice are lowered.  There isn't "proof" available to say that this isn't happening but it can be detected pretty easily if it is.