Don't be pedantic. Do I have to tell you OSX will be 64 bit in a few months?
I'm sorry, but I think your grasp of the technology is pushing people to be pedantic towards you.
Tiger does not have 64-bit APIs.
Leopard *does* -- and will provide 64-bit support on computers that are able to make use of it (C2Ds, possibly G5s, but not G3s, G4s, CSs, or CDs).
But the mini is not now and will not in the future be shipped with a "stripped" OS -- the 64-bit APIs just are not accessible to processors that don't support them. At least, Apple has never shipped "stripped" versions of its operating systems before.
There are only two versions of OSX: the standard OS X and the Server version. The version of OS X on install DVDs is identical to the version on retail boxes with the exception that the install discs are locked to only allow installation on the hardware they were sold with.
When 64-bit apps do ship, it's pretty unlikely that there will be any for some time that "only run" on 64-bit hardware. Rather, they're likely to be designed with forks so that they run on both 32 and 64 bit hardware.
Besides that, it's going to be some time before there's going to be software that most Mac Mini users use that have significant 64-bit usage... it's going to trickle down from compute intensive apps over a relatively long time. Think about it this way. How long have 64-bit processors been shipping on computers (of all OS brands)? Several years. And yet there's almost no need for 64-bit either on Windows or in OS X. This is going to be a very slow transition.