The main point, however, is that your framework automatically ignores anything that adds to every PC and dismisses it as irrelevant and disallows it from consideration.
It's not disallowed from consideration. It's disallowed from consideration
solely for the purpose of solving this problem by itself. Because, for the reasons I gave and you ignored, it can't actually do what you want it to do
by itself. Not theoretically, and not practically.
Letting characters fly with
any DC Acrobatics or Athletics check would never be acceptable to people actually playing the game (practical failure), and would trivialize
both the Wizard's class features AND the Fighter's class features (theoretical failure.)
If your problem is that the gap is large, use that argument. "The extra proficiency doesn't matter because it is tiny next to what wizards do" is a different argument than "adding a proficiency to both wizards and fighters doesn't matter, because anything added to both cannot help fix the gap".
I'm holding the second argument is wrong. If you go and change your position to the first argument? Then you aren't addressing my objection, you are moving the goal posts.
I see your objection as irrelevant, because it's the ultimate conclusion--doing X thing, on its own, is not adequate to solve the problem--regardless of the reason
why that is true.
Like, this is like saying that the fact that Wiles' proof of Fermat's Last Theorem had to be changed on a critical part, means that the fundamental argument is "moving the goalposts." It's not. If there's a
different reason why the fundamental claim at issue is true, what does it matter
which reason one argues? Who cares if you can't ride from London
through Oxford to Cardiff, if you can instead ride from London
through Chichester to Cardiff? It's getting to Cardiff that matters, not the road taken to get there.
Separately, however,
with the theoretical and practical limits forced upon 5e's design, I still maintain that "adding proficiency to both wizards and fighters doesn't matter, because anything added to both cannot
individually fix the gap." The "individually" you left out is the critical bit. It's not that it cannot help in ANY way, because I already granted you that. But by itself,
with 5e's design limitations, it's not enough--and never will be. Your argument requires that we be able to add to both things
without limit. If there's a limit beyond which we cannot add things for any reason, whatever that reason may be, then it is quite possible that we reach that limit and a gap remains. That's exactly what I've argued to you.
"You can make the relative gap between two constants arbitrarily small by adding the right value to both of them" requires that you be able to add arbitrarily large values. But if there's a maximum you can add, the statement is false. E.g., if you can only add +1000 maximum, and the two numbers are 10 and 500, then you can certainly make the gap
much better (10/500 = 0.002, 1010/1500 = 0.67333...), but you cannot get it smaller than 0.9--not even close. 0.67333... is in fact the best you can possibly do.
Further, as the numbers show, you would need to add more than four times that much (4400) to make it work: 4410/4900 = 0.9 exactly. That means adding over 440 times the original power value contributed by the Fighter class, or (as I said) reducing the contribution
from being a Fighter to triviality. Now, perhaps the starting gap is smaller, say by a factor of ten (50 vs 10, as opposed to 500 vs 10.) You'd still need to add a ton: 351 added to both values, a "mere" 31.5 times what the Fighter itself brings. Even if we make it a measly 20 vs 10, meaning the Wizard is
merely twice as powerful as the Fighter, we would still need to add +80 to both values, since 90/100 = 0.9 exactly, which means the Fighter's original contributions are now only one-ninth of its total prowess. For one final example, let's drop it to only W=15, F=10--Wizards being only 50% stronger. To get that to line up, we must increase both numbers by +35--meaning the contribution from the Fighter's own class is still less than a quarter (10/45 = 22.22...%) Being a Fighter is effectively irrelevant for what you're capable of--and being a Wizard isn't much better at only three-tenths contributed by class itself (15/50 = 3/10.)
If we are limited in how much we can add, it is in fact quite possible that we cannot bridge the gap solely through adding an equal amount to both numbers. Even if it
is possible, unless the gap is already quite small to begin with, adding enough to trivialize the gap, for a
very generous definition of "trivialize" (making Wizard no more than 10% stronger than Fighter) means we must necessarily trivialize the contribution coming from the Fighter itself.
In a vacuum, where no such practical considerations and theoretical limits apply, yes, adding to both things could work. But we don't have that kind of liberty with the design of D&D, be it 5e or any other edition.
