Toby:

Yes, in some sense the idea of Turing computation is a kind of physical principle in that no well defined process we know of is not Turing computable (for other readers: this includes chaotic systems and quantum systems as the wave function is computable... with great difficulty in some cases).

Actually, if you built P and it really was very trivial, then I could get my simple Turing machine to compute a quantum level simulation of your P implementation with far less than 3^^^3 bits of extra information. Thus, if your bound really only kicks in at 3^^^3 bits, then (within currently accepted quantum physics) no trivial physical implementation of P can be possible.

Anyway, as you can't specify P in just a few simple states and symbols, I do not consider it to be an acceptable reference machine (for strict theory purposes at least).