Can you go into Boromir's Knot and Hegel's A-N-C?
I have to admit this is far beyond my philosophical knowledge, just have too many negative thetan particles I need Tom Cruise to cleanse me of before I can get into those depths you're swimming in...
Well, I don't believe in coincidences so, the fact that the article presents a triadic structure to History is likely akin to why Hegel formulation of the procession of Geist (Spirit) is also triadic. That is, that when self-consciousness turns inward to look at itself, the result tends to look triadic.
As for the exact formulation, we could likely "fit" the Abstract to something like the article's conception of the Unicorn time-period. The idea here is that we start somewhere, somewhere not definitive and totally articulated. In fact, the article does sort of say this explicitly, but labels it a concrete, which I'd disagree with, I think. The abstraction is the formulation of language, to borrow Lacan's terms, it's the laying of a Symbolic and Imaginary (that is, an ordering built on symbolism, or representations, like language, and imaginary, not just in the sense of being of the mind, but also being of the image, that is, of appearance) onto and into The Real (we can say this is sort of just the material substrate of all things, or the pure facts of the matter).
So, here we start in the laying of the Symbolic and Imaginary orders, we are still in the phase of the Abstraction, we are looking not at what The Real is, in-itself, we are looking to abstract away from that, very specifically, to see what The Real could be. In the next phase though, we introduce (to use Hegel's term) the Negative. So, we recognize the disconnect between the abstraction (the Symbolic and or Imaginary) and The Real, so we start introducing the negation of that abstract thought. What did we "lose" in abstraction? The negation here doesn't really mean a strict mathematical negative, but rather, it's the move to take what we were doing in the Abstract and do the opposite. So, where the Symbolic might have reigned, we look at the Imaginary and The Real. Or however. In the end, we get the "Concrete" from this, because we've gone from one extreme, to apply the other, and now we have met in the "mediated middle."
So, where (to me) the Unicorn stage is the Abstraction, the Phoenix stage is the tarring with the negative, that is, where you go directly against that which was the defining status of the Unicorn. The Dragon is the mediated outcome, where the abstraction comes back in. There isn't an "end" to this process though, no Fukayama-esque "End of History." The structure goes on, where the "final" Concrete stage is actually just the new starting point for another triadic development. Whatever is, will be mediated; the concrete output of that process is just the new starting point of the next mediation.
To apply this to actual historicity, consider something like mystical thinking, "negated" by the scientific revolution, to the concrete stage we are at now, where science is a ruling order, but people do still (likely rightly) still consider metaphysics. Here we go back to abstracting again, considering how the "facts" of quantum mechanics are nearly unintelligible, so we abstract to math to "understand" them. The dialectic starts again.
If you made it through this giant word salad, I apologize. Hopefully it made some sense.