From what you wrote it seems I have not been clear enough. Let me try again:
I am not comparing the fighter and the magic-user AT ALL. They fill different niches due to the different abilities they wield. And THAT'S ALL. Is that better? So my comparison criterion is: you cannot compare them. In mathematical and formal terms, they are two different types.
So, I have done my part; it's your turn: would you be so kind as to provide a meaningful answer to what you mean. Inb other terms: WHAT'S YOUR POINT? That your MP system is better than slots? Or what?
I stand my "faulty argument": fighter and magic-user are "balanced" by design, and different XP tables; they cannot be compared to each other because they fill different niches; in their different niches they provide different gaming experiences, and different measures of flexibility w.r.t. the "space" of the game; and note that I specifically wrote about FIGHTERS and MAGIC-USERS; not FIGHTERS vs. SPELLCASTERS. Because there are some classes which provide both to some extent, and obviously they do not fully embrace both aspects. Why not? Because of GAME BALANCE.
"Balance" is proved through gaming, trial and error, so no faulty pseudo-logic like you provide gives a straight answer. Gygax provided the basics of the game about 40 years ago, and it has stood the test of time quite well, judging from the existence of this place.
Mathematical game theory is a complex topic which you cannot address in a few posts, and you surely cannot provide a formal analysis for a roleplaying game. Now, if you want to play the game of logic and fallacy, we can go on: I am armed with a PhD in mathematics, so things should prove interesting.
Oh, and your comparison is clearly FAULTY:
That said, a much more apt comparison is a fighter armed with a longbow and a sorcerer with magic missile. Both attacks are limited - the longbow attacks are limited by the number of arrows, and the sorcerer is limited by the amount of spell slots or magic points they have.
Why? Take this situation: a battle which lasts the whole day. A 1st level fighter and a 1st level sorcerer join it. The fighter is really an archer, he stands in the back. He has got 200 arrows planted on the ground, he takes them and fires them. How many magic missiles can the 1st level sorcerer cast? 1? 10? 20? Oh well. (oh, and this fact happened in a recent campaign.)
And why they can't be compared? Because they fill different niches. If I have to go into battle for a full day, I will take a fighter. If I need to charm a king and take control of a kingdom (even at first level!) I will take a magic-user.
Hmmm I have edited this post ten times. Though post. On hindsight, I should not have answered at all. Truth is, I am not trying (nor was before, despite what you said) to "prove" anything (and I am not sure what are you trying to "prove" either.)
I was simply stating some self-evident facts. At least, self-evident to anyone who has played any edition of D&D (with the possible exception of the 4th.) Let's call them axioms.
I also realise the tone looks snarky, when it was meant to be tongue-in-cheek.
So I think I am going to bail out of this "discussion", unless you can provide some meaningful (i.e. true, gaming-related) examples, and not some pseudo-logic-based proof (just so we are both on the same page: I can spot a real proof, when I see one; it's my job, after all.) The above example was clearly faulty, made invalid by an actual case.