After thinking the question over, I think I have an answer:

Is there any substantial evidence, not theoritical, to prove Cartman secretly cares for Kyle or sees him in a romantic light? No- however, there is evidence to back up a very clear fixation- a fixation that has led to others such as the groups above theorizing his feelings to be something...more.

Cartman, as we know, has the tendency to remain determined in the face of something he desires, and him hating Kyle is no exception to this. This hatred has culminated into a very strong desire to get one up on his rival and feel superior, which can lead to Cartman going through...questionable means in order to humiliate his enemy. A good example of this is the three parter Imagination Land, one where Cartman dedicates his time to making sure Kyle sucks his balls, as their previously made bet has stated. There's a lot to be said on the topic of Cartman and sexuality- it's implied during the course of the series that Cartman views sexual instances as a means to demean and humiliate rather than anything intimate as most would. Regardless of motives or not, the sexual implications of it are clear, and contribute to the theories.

This, coupled with Cartman's hateful paranoia of Kyle (1%, The Poor Kid, Doubling Down) lead to a very strong fixation on Cartman's side. While context may explain Cartman's motives, the implications itself are enough to have plenty theorizing about Cartman and Kyle.

After thinking it over for a while, I finally have an answer: Is there any substantial evidence, not theoritical, to prove Cartman secretly cares for Kyle or sees him in a romantic light? No- however, there is evidence to back up a very clear fixation- a fixation that has led to others such as the groups above theorizing his feelings to be something...more.

Cartman, as we know, has the tendency to remain determined in the face of something he desires, and him hating Kyle is no exception to this. This hatred has culminated into a very strong desire to get one up on his rival and feel superior, which can lead to Cartman going through...questionable means in order to humiliate his enemy. A good example of this is the three parter Imagination Land, one where Cartman dedicates his time to making sure Kyle sucks his balls, as their previously made bet had stated. There's a lot to be said on the topic of Cartman and sexuality- it's implied during the course of the series that Cartman views sexual instances as a means to demean and humiliate rather than anything intimate as most would. Regardless of motives or not, the sexual implications of it are clear, and contribute to the theories. Another episode, one that basically defines the source of Cartman's fixation with Kyle, is Smug Alert. It provides the basis as to why, despite hating his rival, Cartman most certainly needs him. Kyle provides angry and overblown outbursts in response to Cartman's bullying. These reactions provide negative attention that not only sustain Cartman's narcissism, but his sadisim as well. This desire for entertainment is a bigger priority than his hatred, and so in turn Cartman saves Kyle. Now although certainly a selfish reason, this does prove without a doubt that Cartman needs Kyle around.

While there are many rivalries in media, very few contain the intensity and fixation that Cartman and Kyle's dynamic have, despite being just a sitcom. This, coupled with the implications of episodes such as Imagination Land, are enough to have plenty theorizing about the two.