Closing the gap from wherever Zelda is to reach the so called "dead zone" for Fox is still approaching.
Approaching is defined as (in this case)
-the act of drawing closer to something
-move towards
Zelda cannot win a range battle a full stage length that much has been established. So she is forced to move towards Fox and close the distance between them to reach Fox's dead zone. That's still approaching because she moved towards him.
Even in the dead zone Fox is not trapped or stopped by any means. He can run away to make more camping space, or he can wait and punish. Zelda can try to chase him but he won't catch her. She can try to follow him and throw out smashes but it will not catch him and he will punish you with whatever is appropriate. It's not that hard to do against a Zelda even if you don't main Fox.
The problem with defining it as "moving forward" is that we've been talking about approaching in terms of "approach options", in other words, ability to move through a portion of the opponent's melee safe zone, and when we talk about Zelda lacking safe approach options, is it that she lacks an ability to move forward safely at considerable range? No, it's that her options for moving through the opponent's melee safe zone are very limited.
The point here is that those two are fundamentally different concepts, and in my expirience it's mainly used for "going through the opponent's melee safe zone", but people often extend it explicitly beyond. Really the issue present is not "what it means", but instead "what are we gonna call it then", because if we refer to both as "approaching", talking about a characters lack of approach options is fundamentally dishonest if they can move forward generally but have difficulty moving through the opponent's melee safe zone.
So, what do you suggest as far as technical terminology?
As for your final paragraph, the stage is not endless, if he continues running away, he's eventually trapped in the dead zone (this also happens when you use high-knockback attacks that push him forward), at that point he's basically done your job for you and your focus is not letting him get past.
It's not that I didn't read it, it's just that it doesn't apply to Zelda. SniperWorm summed it up pretty nicely that you have never taken into account that Marth can actually approach well, whereas Zelda's approach is terrible.
But I never was discussing ability to approach, instead dealing with simply moving forward. If you define approach to include simply moving forward at a range, then Zelda DOES have good approach options, just not when moving through the opponent's melee safe zone.
The projectiles have a dead zone, if the projectile dead zone starts at a farther range then the character's melee safe zone then you only need to move into the character's projectile dead zone, and if you define that as approaching, fine, but don't pretend that Zelda is horrible at getting there in this match-up.
You must also take into account that Marth is fast and can actually approach from the air. Zelda is one of the slowest characters in the game, and she cannot approach from the air or the ground well. That makes her more vulnerable to camping, and that makes her forced approach even more disadvantageous for her.
But I did implicitly, the criteria I used was based purely on the attributes, Marth was only an example, a rhetorical device to establish that you can show whether or not a character really needs to approach.
Also, this is out of pure curiosity, but how much do you use Zelda? And do you place in tournaments?
My tournament placements are not notable and I never claimed otherwise. But that's not really relevant since I'm approaching this from the data-based side anyway,
The sad part about theorycraft is that it's just that. Theory. We can talk about how Zelda can stay near this dead zone and not be forced to approach, but we'd be ignoring two simple things that are outrageously apparent when you try to put this into practice:
1. Zelda's approach options are poor
2. Fox can punish like a crazed maniac
Zelda's approach options are bad. Her grabs are terrible and approaching from the air is near suicide. Dare I say it, the only decent ground moves she has are a Dtilt and smashes. But the thing about her smashes is that are a very terrible approach against anyone, and if Fox decides to play super safe and campy, Zelda won't have many chances to get close to him at all, since she's so slow and Fox will be hopping around everywhere. And when she does get close, what then? What are you going to do? Spam smashes? If you lag for a second, you're going to be punished. Fox can close any gap super quickly and hit you with...well...anything. It's Fox after all.
So you play bait and punish, if he retreats, you follow, otherwise play around with spacing.
I'm not saying that this is ultimately advantageous for Zelda, there's no way that this situation even remotely qualifies as an advantage or even an even match-up. All I'm saying is that this is how Zelda SHOULD play in this match-up.
(of course Zelda/Sheik should just use down-b, but we're not talking about Zelda/Sheik)
This is not theorycraft. This has been proven in actual matches, and I'm sure DM has already vouched that Fox does indeed force Zelda to approach. Hell, even NinjaLink says so.
DM meant "approach" differently then I did, and NinjaLink probably meant the same thing. The problem really comes with the tendency to confuse talk about a character as if "need to move forward" and "need to move through the safe melee zone" are the same thing, and this is an issue of terminology being used imprecisely. Unfortunately, poor terminology is very detrimental to match-up discussion (hence why I defined it explicitly, you may not agree with my usage, but at least you know what I'm saying).
As for theory vs. first-hand knowledge.
Scenario 1:
Theorycrafter: Ok, if you do x, y, and z your opponent will **** you because of a, b, and c attributes.
Expirienced player: That's cool and all but I never do that, I do this instead.
Scenario 2:
Experienced Playerr: I've been using this new tactic and it's so awesome, it's a 0-death against x, y, z characters.
Theorycrafter: No, based on the frame data and hitstun data, you've got a vulnerability here and they'll use it against you.
Basically the relationship is, experienced players provide ideas for what to do based on what has worked for them in tournaments, and theorycrafters use the data to figure out if the tactics actually work. They can also provide tactics derived from the data to the experienced players.
The thing is, theorycrafting is deductive, it doesn't have the problem of induction, HOWEVER, it doesn't always necessarily take every possible tactic into account, which is why experienced players are needed as well.
What's preferable is an experienced player that's good at theorycraft as well, however you'll often find that a number of experienced players are crap at theoretical modeling, so you have separate people. They're best used together.