See, the only thing that makes me doubt the Redbeard/Sherrinford theory is that in TSoT, Mycroft is proving a point, almost smug, when he says to Sherlock "Do you remember Redbeard?", and it doesn't seem the way he'd remind Sherlock of their dead brother.
I know a fair number of people have said this, so I guess I’ll just talk about it. The original meta is here: the basic idea is that Redbeard is a symbolic representation and nickname for The Other One, who I’ll call Sherrinford.
This is only my opinion, but if you don’t think Mycroft would use their dead brother to prove a point about the dangers of sentiment to Sherlock, I really don’t know what characterization of Mycroft you’re using. In fact, I think it’s a little weird to think Mycroft would even bother with bringing up Sherlock’s dead dog, but we’ll get into that later.
I’m the first in line to say that Mycroft adores Sherlock; would do (and is doing) anything for Sherlock; sits around on Christmas wishing his little brother liked him better (ASiB); that the highlight of Mycroft’s year is when Sherlock interacts with him; is a soft smoll about Sherlock - but Mycroft never expresses this affection in a genuine or even healthy manner. Wow, it’s another character on this show who’s really bad with expressing feelings and uses mean and spiteful defense mechanisms to deal. Shocking.
The one time in 5 years that Mycroft is genuine with Sherlock, he doesn’t even look at him when he speaks, and as soon as Sherlock is skeptical, Mycroft makes a joke and takes the sentiment back:
Sherlock demands “What the hell am I supposed to say to that?” and Mycroft is like “Never mind; I only said that because I’m drugged”:
Mycroft backs down about sentiment because Sherlock isn’t receptive, but he backs down really fast - he could’ve easily said “It’s true, you know;” he could’ve said anything, but he made it into a bad joke instead. Because he doesn’t want to handle Sherlock’s rejection. And of course Sherlock isn’t receptive, because Mycroft has been telling him how much he doesn’t care about him in ever conversation they’ve had for the last 5 years.
Don’t get me wrong: I love Mycroft; I think he’s very tragic; I know he adores Sherlock; I’m going to cry when he dies; he’s probably my favorite character. But let’s actually look at the things Mycroft has said to Sherlock - because proving a point with his dead brother is par for the course.