From its wiki page, an Axiom is "a statement or proposition which is regarded as being established, accepted, or self-evidently true". Since that also means it's a set of rules that can create or guide the creation of matter, then the System creator (in this case, SYNAPSE with LFE) is the only one that has the full access to the Axioms as he intends it to be. Anyone else will probably only have suppositions and interpretations about them if they figure them out, that is, if the creator doesn't directly declare the existence of the Axiom and an explanation of it to people, but will be unable to change them (unless they are allowed, which leads to this question: can the creator of a System give ownership of it, or outsource its maintenance, to someone else? Should ask Tsuki about that, since I don't remember having that answered elsewhere). Also, LFX is only one framework someone can use to create their Systems, you can create your own from scratch (though I think I remember reading somewhere it could be more standardized once Systemspace 2.0. is up, don't quote me on that though).