What constitutes a blackhole firewall?
I think it's rather different to what you usually read about. I say this as something of a "relativist". I consider General relativity to be one of the best tested theories we've got, see Clifford M Will's paper http://arxiv.org/abs/1403.7377. However I find the given explanation of Hawking radiation unconvincing. Gravitational time dilation is ignored, virtual particles pop into existence, the negative-energy particles obligingly falls into the black hole, and the other particle manages to escape. But virtual particles are field quanta rather than short-lived real particles. They aren't the same thing as vacuum fluctuations. And there are no negative-any particles.
This firewall is, essentially, high-energy quanta that is needed to break the entanglement.
But we don't see firewalls breaking entanglement in a lab. However what we do see, is gamma ray bursters. And when you look around on the internet, what you also see is Friedwardt Winterberg making comments about the firewall paper he wrote in 2001. There's maybe some kind of priority dispute going on here, see this old Wikipedia article which mentions AMPS but says Winterberg's proposal has priority. Winterberg might have been the one who wrote that, but nevertheless see reference 87 in An Apologia for Firewalls. Anyway, here's his paper: Gamma Ray Bursters and Lorentzian Relativity.
My question is what is the cutoff for the firewall? That is, how energetic could it be? I really would like to see some quantitative/mathematical analysis of the subject in a pedagogical manner, but whenever I try to learn more I get popular articles.
I can't tell you much about the AMPS firewall or the entanglement I'm afraid. But I can sketch out why I think Winterberg's firewall is more or less correct. Take a look at Einstein talking about the speed of light here: "As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable". Light curves because the speed of light is spatially variable. And matter falls down because the speed of light is spatially variable. Not because spacetime is curved. Einstein never said that. Matter falls faster and faster as the "coordinate" speed of light reduces. But if this continues unabated, there would come a crossover. There would come some point when the falling body would be falling faster than the coordinate speed of light at that location. That just can't happen, because of the wave nature of matter. Something's got to give. Things like electrons, neutrons, and protons. The wave nature of matter is not in doubt, and in gamma ray bursts, it would seem that those waves break. Along with all the rules. As for how energetic they are, we are talking about the 100% conversion of matter into energy, so the answer is: very.
Edit: I'm surprised there's no other answers Joshua. In response to your edit, can I add that the article says Polchinski is a co-author of the paper that started it all, but it isn’t true, which is why Winterberg commented. IMHO what is true is that some speculations are promoted via popscience articles that don’t stand up to scrutiny. For example, see this: "In this scenario, colorfully dubbed 'No Drama', the gravitational forces won’t become extreme until she approaches a point inside the black hole called the singularity. There, the gravitational pull will be so much stronger on her feet than on her head that Alice will be spaghettified". Gravitational force at some location relates to the gradient in the coordinate speed of light at that location. At the event horizon it’s zero, and it can’t go lower than that. Also see where it says this: "This radiation results from virtual particle pairs popping out of the quantum vacuum near a black hole". That’s popscience bunk. Virtual particles are field quanta. It’s like you divide an electromagnetic field into abstract chunks, and say each is a virtual photon. Then when the electron and the proton attract one another, they “exchange field” such that the hydrogen atom has very little in the way of a field left. As for towards quantum gravity, when two hydrogen atoms attract one another gravitationally, they don’t exchange field: the resulting field is doubled up, not diminished. And as for this is a terrible blow to general relativity, I think not.