I'm still digesting the proof and translating a bit from the dual case back to the primal in which I most commonly thought about it. I don't think it was a brute force proof in the sense that it combined every possible paper and commentary. It's rather odd because I feel like most of the work on the conjecture was focused on an induction proof based around graph reductions, and this proof avoided those issues entirely by offering a concrete constructive proof of finding a Hamiltonian cycle. Rather, it explicitly selected the edges not in the Hamiltonian cycle, which is in line with previous attempts via the dual.

The "aha" insight for this is actually f**ing wild, it involves a complex valued exponential sum on the edges. I've seen a lot of clever counting arguments before in graph theory but this is the first time I've seen complex roots and annihilating terms like this, the symbolic manipulation tricks in this look like things out of quantum physics. I don't understand where this trick originated, I need to really digest this.