>

Folding is superior
I suggest you further divide category 2 into 2 subcategories:
2a: People who scrunch up a wad held in the hand.
2b: People who wrap paper around their hand, entirely enveloping their hand within a scrunched quantity of paper. >

I refuse to believe 2b exists. no way >

I wasnt aware that 2a exist until this post. But i had seen idubbbz's kickstarter crap video about toilet paper "gloves". (link) >

When you scrunch it up, you get lots of creases in the paper which adds more surface area and structural support.   
Picture a puddle on the ground: If you lay a sheet of paper down on top of the puddle, the sheet will get soaked. If you touch the sheet, your hand will get wet. You would have to pile up several sheets to get to the point where you're not risking getting  your hand wet. Even then, it might soak through or the paper might tear and then you're screwed; after all, even with many sheets piled up, your hand is still only a few mm away from the water.   
Or instead, crumple a ball of paper and drop it right in the centre of the puddle. You can now safely grab the top of the ball of paper and drag it around in the puddle. You're hand stays perfectly dry, and there is considerable distance between your hand and the water. Do this multiple times with the same amount of paper you used in the first scenario, and you've absorbed the whole puddle now with less risk of getting your hand wet. You can press as hard as you want, with less risk of a tear putting you in danger.    
Also, consider that there might be some solid dirt stuck to the floor at the bottom of the puddle.  Dragging a smooth sheet of paper across the dirt isn't going to clean it well. Dragging the rough ball of paper across it will scrape it off much better. >

Unless you have severe diarrhea your asshole is not a puddle. Even if it were a puddle, it would be a puddle of sub-mm depth. Even if you think you're not going to get your hands wet, I hope you still wash them. So, even if you're right the point of sparing some dry tp to touch is meaningless. 
Your asshole does not have the requisite topography to resist cleaning by mostly smooth surfaces with applied pressure. It's not dragging. Dragging is a completely misleading analogy as the pressure applied is way more than gravity. Further, the effective surface area of the scrunched ball over a given arc is definitely less than an actual ball. With pressure, any sulci would press into one another meaning that the scrunching provides close to zero benefit to surface area. It's equivalent to wiping your ass with maybe 10-15 layers of tp folded against one another with the layers aligning opposite the direction you fingers are pressing. I'm not sure how much that is, but it's way less than than the several inches that are possible with sheets of tp. >

Balling/scrunching up, if done somewhat evenly, will give you more material concentrated around the center. When wiping, this gives you extra frilly mini-layers between your hand and the dirty part of your bum, where it matters most. You'd have to waste more toilet paper to increase thickness to prevent accidental ripping or any backsplash water from seeping and reaching your fingers. Speaking of seeping water, completely neat folds will have the water travel the shortest path to reach you. >

Use the most effective surface area flat tp, and wash your hands. >

IMO there are 7 factors to consider

TP as a resource
Time
Frequency of getting shit on your fingers
Properly cleaning your ass
Leaving behind TP
Versatility for TP quality
Plumbing

I'm a folder by habit at this point, having been raised to prioritize the efficient use of TP above all other factors. However, now that I'm an adult with my own income I would say that I've adjusted to prioritize other factors even though I continue to fold. 
It is almost certainly true that folding is more efficient. Folders are counting how many sheets they use, and most often there's a set number of sheets folders use depending on the kind of mess there is to clean (e.g. the messier wetter shits require more sheets. 
The flip side of this is that it takes time to count out your sheets and fold them. Wadding is therefore a more efficient use of time. If you want to finish wiping in under 30 seconds wadding is a must, but even if you're not in a rush and just want a thorough cleaning as quickly as feasible wadding is still probably going to save you time. 
While there's a clear winner in the first two categories, that is not so for the next two. Given enough toilet paper folding does a very good job of keeping your fingers clean. The distribution of paper is even, so given an equivalent amount of paper you're less like to poke a hole through a thin spot when folding. OTOH, the sheer volume of paper that wadders tend to use make it less likely that any spots where fingers are applying pressure is as thin as your average folders fold. There's also just less surface area, generally speaking, when folding, which could lead to getting shit not only on your fingers but the side of your hand. Here I think wadding edges folding but a tightwad wadder is almost certainly worse off. 
In terms of properly cleaning ones ass, I think we would need to see some studies to see which one comes out ahead. Hasty wadders probably leave a lot of shit behind, but given the same amount of time spent wiping wadders probably do better by getting more wipes in. 
One thing that it is definitely superior about folding is that it leaves less toilet paper behind. Since that paper probably has shit on it I place a high value on this, and it's probably the main reason I've never really embraced wadding. 
Having said that, even I will go for a good wad when dealing with low quality TP. Those huge sheetless rolls of industrial TP typically find at gas stations are worthless when neatly folded. Since I'm not as practiced at wadding, I'm sure that I fail to get my ass cleaned as well as I would were I wadding consistently. But at home I spring for the good stuff, and when working with a nice soft two-ply, I've got to fold in order to avoid leaving behind a dingleberry farm. 
But this all starts with good plumbing. If you've got bad plumbing, you might be having to throw TP into a waste basket. In my experience a folded piece takes up less space and is easier to fold one last time so that no shit can be seen. But if the plumbing is good enough to withstand cheap TP but not the good stuff then we're back to wadding that trash up. 
One last wild card here: finishing with a wipey. Again if the plumbing can take it then that extra wipe sweeps away just about anything leftover, whether shit or paper. 
Given everything, I think the versatility and time saving features of wadding even the playing field, especially if you can mitigate the downsides with a wipey. 
Not saying it's better, but you definitely can't say that folding is the only proper way of wiping one's ass, or that it is certainly superior to wadding. >

This is the type of quality discussion I subbed for. >

Depends on the toilet paper quality. My family used Scott’s single ply tp growing up and there was no way to prevent soak through while using a reasonable amount of tp without scrunching. Now I buy the great value knock off charmin and 2-3 sheets folded is more than sufficient and wadding would be pointless and wasteful. >

I feel like when you scrunch it up, there is less of a chance of you finger poking through.  
Also, less surface area might be better if you are forced to use low quality toilet paper, AKA sandpaper. >

Let's say we are using an equal amount of toilet paper, I can't imagine balling it up can make less chance of a finger poking through, unless you are balling it so tightly that it would seem less usefull for it's intended purpose. 
I guess to compare that, I'd need to know how many squares of toilet paper we are talking about here. I'm thinking folders use 4 or less squares... do scrunchers use significantly more than that?
If so, I guess with a much larger amount balling it up would make for lower chances of a finger getting through the larger amount of material. >

It's a safe bet that scrunchers use more. Like you said, 4 squares folded is adequate for a typical swipe, but the same  amount scrunched would be useless. >

Be civilized and fold your damn paper >

Firstly, the "Less surface" area is highly suspect given how maleable toilet paper is.  That logic fits if TP where steel, but not really when it forms to the shape of what it is pressed against regardless.
And...that's kinda the end of the story.  What matters is the shape of your hand as you press against le' assshole', not the shape of the TP prior to being pressed.  These "air pockets" you describe simply do not exist unless you're not strong enough to overcome the basically non-existent resistance provided by TP. >

I would assume that either way, balling or folding, a person would be using the same amount of surface area of their hand, so I am assuming it is like wiping a counter with an amount of paper towel. 
If you have two spills on a counter, and you take two amounts of paper towel, one for each hand, and in one hand you fold a few pieces of paper towel, and in the other you ball up the same amount of paper towel, my theory would be the folded paper towel would cover more area faster, and leave less behind. >

Why not use both techniques? >

Fold it and then scrunch it? >

Folder here. Always have been, always will be. It is the only way. >

IT HAS ALWAYS BOGGLED MY MIND THAT ANY SENTIENT AND CONCIOUS HUMAN BEING BALLS UP THEIR TOILET PAPER. 
Fucking cavemen. Do you smash pizza against your face and hope it gets into your mouth?  Do you put your legs through your shirt hole and pull it all the way up your body before putting your arms in? Do you buy a new car when your current one runs out of gas?
No, you don't do those things, because you are not insane.
So why would you ball your toilet paper? >

There are only 2 types of toilet paper users

those that fold
those that have poop on their butts >

With decent toilet roll, getting 6-8 sheets, folding in half and wiping... perfect way of wiping your arse. Never tried screwing up paper. Didn't know it was a thing. Doubt I'd try it. >

Going to argue against myself here with the same argument someone else had.... is it possible there could be more wiping ability with scrunched up paper? Could the increased balling effect make it less likely fingers pop through? >

Scrunching creates liitle pockets and crevices which capture more poo with the wipe.  When folding, you are more likely to just push it around smearing poo further around your bum, and this is especially bad for hairy bum people. >

The truly superior method is to not use TP in the first place* and get a bidet. You can get them for like 40 bucks on Amazon that will install on basically any toilet 
*You would still use a square or two to dry off afterwards, but pressurized water is much better at cleaning your bunghole than thin paper. No risk of Dookie on your hands either.
There's a reason "pressure washer" videos are so "satisfying". 
I can't even go at work anymore because of their lack of bidet. I wait until I get home. >

Since you didn't specify the use case very exactly: scrunched up paper offers way more absorbent material between your butt and your fingers, so any liquid doesn't soak straight onto your hand. Not so useful for wiping after number 2, usually, I agree. Though the little folds can add some cleaning, maybe.
Very useful when wiping after peeing if you have a heavy period, and don't want to get drenched in blood for obvious reasons. >

You are both correct and incorrect at the same time.
Imagine that, through some unfortunate event, you got fecal matter on your forearm, or perhaps on your cheek. Are you seriously telling me that the appropriate thing to debate is whether to clean it with a dry paper towel that is balled up, or a dry paper towel that is folded, and we should just ignore the idea of using water to clean it off? Ludicrous!
Once you have a bidet installed, you will find that folding toilet paper dries your bum much more effectively that balling up your TP. So in that sense you are correct, but I hope I have changed your view about ignoring that third option. >

If I got fecal matter on my arm, I wouldn't just rinse it off with water. I'm scrubbing that shit off with soap. >

No way.    
All you're doing is smearing.   
Making a peanut butter sandwich with a knife instead of cleaning peanut butter off the counter with a sponge.
It's all about the nooks and crannies when it comes to removing material >

Depends on the angle
Much easier to clean up peanut butter with a putty knife than a sponge (particularly a dry sponge) >

Paper doesn't have a rigid edge.   So, no. >

How are you going to say that folding smears like a knife, but can't do the opposite? >

Because a flat surface and a rigid edge are different things. >

sponges are also flat >

Still all about the NOOKS AND CRANNIES when removing material. >

Except not really because a putty knife cleans faster and more efficiently
Even if you are talking about "rigid edges", flexible silicone spatulas and bench scrapers are far more common in gathering up sticky and smudgy messes, like for peanut butter in cooking >

Balling up done correctly creates a bell curves like distribution of TP thickness. Assuming the same amount is used, this either 1) makes it thicker where your hand is applying pressure and is most likely to break through, and/or 2) creates a larger surface area to better prevent incidental contact. The differences between tp and paper towels is that with tp you are not wiping a flat surface like a counter top, you prioritize not touching what you are wiping more, and you are not absorbing but rather scraping off the substance. For these reasons, having an inconsistent thickness can provide more tp where your hand is wiping and providing more surface area in case of incidental contact (side of your hand) >

Best argument yet if you can explain how it can possibly be true...
Do you scrunch it in a specific way every time to make sure it always works the right way? I had imagined it was just absentminded tearing and balling in a hasty fashion. >

Not really it’s been 20+ years of unintentionally trial and error so it’s kind a just habit. I don’t think of exactly how I put my shirt on every morning, I’ve just kinda figured out the best way and don’t really think about it. Naturally tho any sphere will be flattened into a rough bell curve shape and the size/compaction before flattening is just more of fine tuning.
It’s especially effective for single ply tp when you really need more layers at the point of pressure but it doesn’t really matter how many ply protect grazing contact. 
An additional advantage is that the fold/layers formed at the surface of a ball provide a better scraping surface than flat tp >

Yeah if you’re trying to shit on your fingers >

i scrunch it up and stuff it up my butthole, utilizing maximum surface area. i proceed to twist it around, and when i pull it out, it will have cleaned out everything. this is why i only wipe once. >

Balling up the toilet paper, when done with a correct application of chaos, actually produces more surface area than a flat plane of toilet paper - much like the folds in a brain. It also provides pockets and crevices for the storage of fecal matter that you are trying to remove.
So if you are talking about efficiency of toilet paper usage, you are generally correct that folding will do more with less. Unless you have experienced a bad case of "mudbutt", in which case your methodology needs to be amended to remove the mass of mess before you can get down to some refined cleaning with planar wipes.
I generally fold, but there are times when I have to switch strategies  go into chaos tp ball mode. We won't discuss when I've had to shower afterwards. Those were dark days and I've since changed my diet. :) >

Who tf balls up toilet paper?
Also I roll mine around my hand. >

I do use bidet but use toilet paper too to soak water.
and I found crunching toilet paper in a ball or best a rugby ball shape ( prolate speheroid) roughly speaking, is best.. because crunching toilet paper gets rough folds. and if you use on just shitted holes, then the uneven creases will do the job better >

Not when you’re angry it’s not... >

It really depends on the factors if you wipe top to bottom or bottom to top, along if you scrub with each piece of toilet paper or only wipe once per wad/section. 
Now if you’ve graduated to the bottom to top technique or mastered the scrub style, the wad/ball may be superior depending on your technique. So overall there’s so many factors to consider that it’s not a simple answer of folded vs wadded. >

Scrunching for last bits of a wet job. >


Scrunched definitely creates more surface area.
It provides a better grip so more force can be used. I.e. f=mxd so m=f/d so greater force with the same distance acquires a greater mass.
It can folded in on itself 3 times at a minimum before it goes in the pot.
The only downside is a brown thumb occasionally, which can be cleaned during the washing ritual. >

I’ve been spending most my life doing just that, and I’ll keep spending most my life doing just that >

You have the wrong number