Quote - but I seem to remember 64-bit OS's don't work with 16 bit code.

With mainframes the story is rather different, the old ferrite core memory IBM /360 was a 64 bit machine.
Burroughs machines were 48 bit and the machine language and instructions was Algol and not Assembler!. Compatibility between different models was on base on the source code and not the executable code. For running an application you have to compile, link and run all in a batch process, even you assigned all the resources that will be used by your application.
The question of 16 bit or 32 or 64 bit code is the result of the evolution of microprocessors that started with 4 bits and now we have 64 bit, 40 years after the mainframes had it!!
As the executable code is all not the same for any new microprocessor and the compabibilty is not based on source code, secret, profit, copyrights, even more profits, the only way to a new processor or OS to be compatible with old applications, the hardware needs some switching code mechanism and the OS must support this.
Without the compatibilty the sales of new computers would be very limited. Nobody would purchase a new computer if all the software that he has cannot run on it. It's not only a question of all the money needed to purchase all the new required software, many times the software that you have doesn't exist a newer version for a newer hardware.
Imagine if Poser7 would not be able to run on 64 bit  computers, you would never have purchased an AMD64 or Intel64 until the day where Poser 8 would be released, and who knows when it will be, neither you don't know when the first patch of Poser7 will be!

Stupidity also evolves!