Axioms are written in assembly, the file extension for which is .z80 or .asm
You use an assembler and possibly another tool to assemble them and turn them into .8xp files, which can be sent to your calc.
If you don't know ASM and want to learn, TI-83+ Assembly in 28 Days or Hot_Dog's ASM for the absolute beginner are recommended
Edit: http://future_history.freehostia.com/Files/Resources/ASM/ASMin28Days/lesson/toc.html is a link for the first

True, unless you have been doing BASIC for an incredible amount of time and just can't get used to low level stuff. In my case, I failed with anything other than TI-BASIC and Axe. I even spent months trying to learn ASM, split in 3 attempts (May-June 2004, Feb-March 2005 and somewhere else in 2005) as well as 68k C (through Technoplaza and another tutorial, in french, by Squale92). The thing with ASM, though, apparently, is that it's incredibly hard to learn for many people, but once you grasp the concept, it's not too hard to code. It just takes a long while to code large games since one single command is like 10 or 20 lines of code. One idea I have is to know Axe first, though, because you can understand the concept about pointers or RAM a bit easier afterward.

You get used to it eventually.
Back on topic, (well, sort of), I find it amusing that an Axiom's .org statement has value $C0DE

That's not a .org statement, that's just a unique header name so you don't accidentally import an appvar that's not an Axiom because no further checks are done after that.  Its more like the $6DBB used by the TI-OS, which is the AsmPrgm token.  If I do decide to merge Axioms with external subs that would kind of invalidate its use for the contest because there would be no way to tell if the generated code was written in Axe or assembly just like the executables.  But I suppose if the source for the Axiom were released, it could be compiled using Axe to prove it.

There are other technical limitations though like a size limit for each routine.  I think to avoid those, I could have the Axioms just be source code rather than assembled code so they can compile just like regular source code.  But the other thing I dislike about merging the Axioms with external subs is that they are not easily modifiable.  If you need to make a small change to the Axiom like to change the dimensions of a tile mapper or something, you would have to have the original source code to recompile it.  But if you have to do that, I might as well just have a separate way to do external subs with the actual program itself.

Who suggested merging Axioms w/external subs? Not me.
They'd definitely be better separate, IMO, since they serve different purposes. Axe Axioms are considered Axioms, not external subs, right?

Edit: http://future_history.freehostia.com/Files/Resources/ASM/ASMin28Days/lesson/toc.html is a link for the first

Is that a link to my site I see?

THey used to be hosted by Dragon-Fire programming team but the server shutted down a long while ago. The URL was extremly hard to remember too, especially that it used a ~